Four people were subsequently removed from the hearing, where they had been in the public gallery holding up pictures.
Mr Johnson told the inquiry: "I am deeply sorry for the pain and loss and suffering."
One protester held up a poster reading: "The dead can't hear your apologies."
Mr Johnson went on to say he hoped the inquiry was able to "get answers to those very difficult questions" victims and their families are "rightly asking".
Asked to confirm whether he believed COVID sufferers and bereaved families were "entitled to full and rigorous scrutiny" of him, Mr Johnson said: "Of course."
Boris Johnson COVID evidence live: Former PM apologises to victims as he begins marathon evidence session at inquiry
The evidence session also heard:
• Mr Johnson believed there were "strong arguments against going too early into lockdown" in March including "behavioural fatigue" and "the risk of bounce back".
• People in Number 10 got "very frazzled" as COVID "kept coming in wave after wave"
• He may have only read Scientific Advisory Group for Emergencies (Sage) minutes "once or twice".
Watch a Sky News special on Johnson's day of evidence at 9pm
The former prime minister was speaking on the first day of his appearance at the official COVID inquiry he set up in order to learn the lessons of the pandemic for the future.
He is the inquiry's most highly anticipated witness and follows on from fellow politicians including former health secretary Matt Hancock, former deputy prime minister Dominic Raab and Levelling Up Secretary Michael Gove.
Mr Johnson will be questioned for two days about decisions he made which took the country into three national lockdowns.
'Should have done differently? Unquestionably'
The former prime minister told Hugo Keith KC, lead counsel for the inquiry, that "unquestionably" mistakes were made by his government during the pandemic, adding that he took "responsibility for all the decisions that we made".
Pressed by Mr Keith KC on what mistakes he felt were made, Mr Johnson cited communications and the different messaging coming from the different governments in the UK.
Mr Johnson also said he took responsibility for the speed of the government's response to the pandemic. the lockdown decisions and their timeliness, the circulation of the virus in the residential care sector and the Eat Out to Help Out scheme.
He said he acknowledged that "so many people suffered, so many people lost their lives", the government was "doing our best at the time, given what we knew, given the information I had available to me at the time, I think we did our level best".
"Were there things that we should have done differently? Unquestionably."
The inquiry, which is now examining decision-making and political governance, began with Baroness Hallett raising issue with the briefings ahead of Mr Johnson's appearance, arguing that a leak "undermines the inquiry's ability to do its job fairly, effectively and independently".
Johnson questioned on 5,000 missing WhatsApps
In the days leading up the inquiry there were reports anticipating Mr Johnson's apology and the fact that not all of his WhatsApps would be made available to the inquiry - with about 5,000 messages on his phone from January 30, 2020 to June 2020 missing.
Mr Johnson said he did not know the "exact reason" they were not located, but said it was" something to do with the app going down and then coming up again, but somehow automatically erasing all the things between that date when it went down and the moment when it was last backed up".
Mr Keith said a technical report provided by the former prime minister's solicitors suggested there may have been a factory reset on the phone at the end of January 2020 followed by an attempt to reinstall its contents months later in June - something Mr Johnson said he did not remember.
"Can I, for the avoidance of doubt, make it absolutely clear I haven't removed any WhatsApps from my phone and I've given you everything that I think you need?" he said.
As well as politicians appearing before the inquiry, other figures that have given evidence include top scientists at the time - including Chief Medical Officer Chris Whitty and former chief scientific adviser Sir Patrick Vallance - and Mr Johnson's former chief adviser Dominic Cummings.
Mr Cummings has previously given evidence to the inquiry in which he described Downing Street as in a state of "complete chaos" and claimed that he urged Mr Johnson to remove Mr Hancock - whom he claimed "lied his way" through the pandemic.
Read more:
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Government had 'challenging and competing characters'
Mr Keith told Mr Johnson that the WhatsApp messages that have been shown to the inquiry "paints an appalling picture, not all the time but at times, of incompetence and disarray".
Mr Johnson argued that plenty of successful governments have "challenging and competing characters whose views about each other might not be fit to print but who get a lot done".
Asked about comments he made in which he called Mr Hancock "totally f***** useless", Mr Johnson replied: "My job was not uncritically to accept that everything we were doing was good. I do think that the country as a whole had notable achievements during the crisis."
He admitted that while he was aware Mr Cummings had a "low opinion" of the health secretary, he thought Mr Hancock "worked very hard, he had defects, but I thought that he was doing his best in very difficult circumstances".

The incident is alleged to have happened at Christ The King School in Islington, north London.
A statement from Islington Council on X, formerly known as Twitter, said: "We are aware of these reports and are taking them seriously. We're working with the school to investigate."
On Tuesday, Victoria Azubuike, posting on the social media platform, said: "I can't believe I'm writing these words. My 10-year-old nephew was punched multiple times by his teacher… at Christ The King School in Islington.
"The 10-year-old had to go to A&E immediately after reporting severe pains across his chest and back areas."
Sky News has contacted the school, which said it was not releasing a statement "at this time".

The Dutch edition of Omid Scobie's book Endgame claimed they raised "concerns" during family discussions about how dark the skin of the Duke and Duchess of Sussex's son might be before he was born.
Many national newspapers followed the lead of broadcaster Piers Morgan, who named the two royal figures on his TalkTV show last week.
Buckingham Palace is considering whether it should take action over the publication. When the unsubstantiated allegation first surfaced two years ago, the palace described it as fiction.

The new pictures - which many will see as a show of unity - were taken as the Royal Family welcomed ambassadors, high commissioners and other diplomats to Buckingham Palace.
The white-tie event is held every year and sees more than 500 members of the Diplomatic Corps hosted in the palace's state rooms.
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Endgame 'error' won't help Harry and Meghan
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The Queen wore the Girls of Great Britain and Ireland Tiara, while Kate dazzled in the Lover's Knot Tiara - a diamond and pearl-encrusted headpiece made in 1914, which was often worn by Diana, Princess of Wales.
It was given to Diana by the late Queen as a wedding gift in 1981.
The diplomatic reception is usually held every December, but there was a pause in 2020 and 2021 due to the COVID pandemic.

The report comes after alleged victims of the drug had their legal challenge against the manufacturer and the Department of Health struck out by a judge in May this year.
Primodos was a hormone-packed pill and, in the 1960s and 70s, GPs gave it to women to test whether they were pregnant.
This new report produced by a Swedish professor of pharmacology and toxicology claims the drug could cause damage to the foetus in similar ways to abortion drugs.
Bengt Danielsson, who has worked in the field of drug safety and teratology for 35 years, spent two years analysing data from studies on Hormone Pregnancy Tests (HPTs), concluding the drug had the potential to cause a range of congenital problems, such as shortened limbs, skeletal malformations, and cardiovascular defects.
Campaigners plan to raise the findings in a meeting on Wednesday with the minister for patient safety, who agreed to look again at the issue following a heated debate in parliament in September.
Previously, the government had apologised for letting down the alleged victims of Primodos, before siding with the manufacturer to block their legal claim.
HPTs were sold under different trademarks, the most prevalent in the UK was called Primodos.
Unlike the urine pregnancy test that was developed later, these were two pills that were handed out by GPs to be taken 12 hours apart.
This would give the woman a rise in hormones, followed by a rapid decline, mimicking the end of the menstrual cycle, which in non-pregnant women would trigger a menstruation bleed.
If a pregnant woman used HPTs, she would generally have high levels of pregnancy-induced progesterone, which maintains pregnancy normally and there is no bleed - and this is how the woman knows she is pregnant.
However, Professor Danielsson's study suggests that HPTs have potential for a third outcome - to initiate a failed abortion process, resulting in uterine contractions and bleeding in some pregnant women - most likely those women with naturally lower progesterone levels when pregnant.
He argues the hormone spike from the pregnancy test could give women with low progesterone levels uterine cramping, resulting in the womb attempting to expel the uterine lining with the living embryo.
His paper suggests this also decreases blood flow to the embryo, starving tissues of oxygen (hypoxia) and when the oxygen returns, this can also impact recently formed blood vessels within the embryo (so called "vascular disruption") which can damage whatever is developing at the time.
This could range from shortened limbs to hand defects - to damaged internal organs such as the heart and brain.
Professor Danielsson argues this "vascular disruption" is the same mechanism that can occur with the morning-after pill, Misoprostol, if it somehow fails to abort the embryo.
He says his hypothesis is supported by numerous factors, not least results in a human clinical trial in Australia, where a proportion of women who used HPTs showed "spotting" and signs similar to an early threatened miscarriage.
He also notes that the types of malformations seen in alleged HPT victims are near identical to those associated with Misoprostol.
He told Sky News: "Depending when in pregnancy of having this hypoxia, the oxygen deficiency event, that also tells you what type of malformations you would get.
"So, if it's early on, it would be more severe, for example an amputation of the arm. Later, it might be the branches of the latest developed vessels such as the fingers."
Professor Danielsson is also critical of an expert working group EWG commissioned by the British government, which concluded in 2017 that they could not find evidence to prove a causal association between hormone pregnancy tests and malformations in the embryo.
He argues that the datasets within the annexes of their report show the association exists.
He said: "One of the annexes to the EWG report, highlighted that HPTs may cause embryonic hypoxia by a similar mechanism as Misoprostol and that several types of malformations associated with HPTs were of the same 'vascular disruption' type as shown for Misoprostol."
He added: "Two extensive epidemiogical investigations on HPT-associated malformations, based on totally different populations and different methods, were presented in annexes to the EWG report.
"Both showed consistency in increases of several specific defects, however this important aspect was neglected."
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Charles Feeny, the barrister who represented the alleged victims of Primodos, said: "It was like they (the EWG) had the pieces of a jigsaw on the table - and it's a complicated jigsaw and you've got to put the pieces together carefully and they just weren't able to do that.
"What Bengt Danielsson's done is put all the pieces together - and there you can see that clear picture.
"Hormone Pregnancy tests did cause malformations, but they caused them in a small group of women, the women who were susceptible to it, probably because they had low progesterone levels, even though they were pregnant."
Mr Feeny also believes the drug would have caused some women to have abortions.
The High Court judgment in May dismissed the legal case in the UK. Previous litigation against Schering, which is now owned by German manufacturer Bayer, also failed in 1982 when the claimants' legal team decided to discontinue on the grounds that there was no realistic possibility of success.
Bayer told Sky News: "Since the discontinuation of the legal action in 1982, Bayer maintains that no significant new scientific knowledge has been produced which would call into question the validity of the previous assessment of there being no link between the use of Primodos and the occurrence of such congenital anomalies.
"In 2017, the Expert Working Group of the UK's Commission on Human Medicines published a detailed report concluding that the available scientific data from a variety of scientific disciplines did not support the existence of a causal relationship between the use of sex hormones in pregnancy and an increased incidence of congenital anomalies in the new-born or of other adverse outcomes such as miscarriage.
"The Committee for Medicinal Products for Human Use of the European Medicines Agency supported that conclusion."

Damion Johnson, 53, had known 71-year-old John Wainwright for 27 years - and moved in with him into a flat in Birmingham in 2015 as his registered carer.
The pair were described by a friend as having a "strong friendship" and Johnson described Mr Wainwright as a father figure.
Derby Crown Court heard when Mr Wainwright died in September 2018, Johnson was "overcome by grief".
On 25 September 2018, he ordered a chest freezer measuring around two feet by three feet, costing £462.
Mr Wainwright's body was discovered almost two years later in August 2020 in the freezer on the premises of a skip company.
Johnson told relatives and friends that Mr Wainwright had died and that the funeral had already taken place, but did not inform the emergency services or obtain a death certificate.
At one stage, a friend of Mr Wainwright stayed at the flat while the pensioner's body was stored in the freezer in the same room.
Prosecutor Darron Whitehead said: "The defendant says he was not thinking rationally and was not ready to let go of Mr Wainwright. As time passed, he had been unable to inform the authorities."
In December 2019, the defendant was arrested for unrelated matters, and while police did not search the property, they barred Johnson from returning.
The flat was boarded up on 6 December with the freezer unplugged inside.
Several people later attended the flat to carry out safety checks and noted a strong smell which they described as "horrendous" and "unbearable".
On 21 August 2020, a removal team took the freezer away and mistook the odour for rotting food - before Mr Wainwright's body was found by staff at Budget Skips Services Ltd in Exhall, Warwickshire.
A post-mortem examination five days later noted signs of blunt force trauma, but Mr Whitehead said: "It was not possible to confirm or exclude natural disease as a cause or contributor to death."
The prosecutor said from September 2018 to May 2020, Johnson also used Mr Wainwright's bank card to buy goods and withdraw cash worth £17,000 and made 11 transfers to his bank account worth an additional £2,475.
The 53-year-old was sentenced to two years in prison on Tuesday after previously pleading guilty to preventing the lawful and decent burial of the body of Mr Wainwright, as well as three counts of fraud.
He was given concurrent six-month sentences for each of the latter charges.
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Raglan Ashton, mitigating, said Johnson had previously worked as a carer and a healthcare assistant at the Royal Derby Hospital and had an "informal agreement" with Mr Wainwright that whoever died first would still be able to access the funds which were in a joint account.
He said: "Perhaps one can understand that if one sees it against that background, perhaps it was not a callous act but an act, clearly inappropriate, of someone who was finding it very hard to come to terms with the death of Mr Wainwright."
Jailing Johnson, of Sun Street, Derby, Judge Shaun Smith KC said preventing a burial was an "unusual offence" but that he was "not suggesting at all" that the defendant had any involvement in Mr Wainwright's death.
He said: "Had you accepted his death and gone about it in a normal way, he would have received a good and decent burial.
"That was not what you did. You bought a chest freezer, a deliberate act on your part. You knew what you were going to do.
"Everything you did facilitated the hiding of that body. Nothing you did contributed to it being found.
"This is an offence which is so serious that the only appropriate punishment can be achieved by immediate custody."