Boris Johnson has backed his key adviser Dominic Cummings, amid a row over the aide’s travel during lockdown.
The PM said he believed Mr Cummings had "no alternative" but to travel from London to the North East for childcare "when both he and his wife were about to be incapacitated by coronavirus".
"In every respect, he has acted responsibly, legally and with integrity," Mr Johnson said.
It follows calls from several Tory MPs for Mr Cummings' resignation.
Labour leader Sir Keir Starmer said Mr Johnson's decision to take no action against Mr Cummings was "an insult to sacrifices made by the British people".
Leaving Downing Street after about six hours in Number 10 on Sunday, Mr Cummings refused to answer questions.
Meanwhile, Mr Johnson said parents and teachers should prepare for the phased reopening of schools in England to start on 1 June as planned.
He also announced that a further 118 people had died with coronavirus in the UK, across all settings, bringing the total to 36,793.
On Saturday, Mr Cummings and the government had said he acted "reasonably and legally" in response to the original claims that he drove 260 miles from London to County Durham with his wife, who had coronavirus symptoms.
But, speaking at Downing Street's daily coronavirus briefing, Mr Johnson called "some" of the claims "palpably false".
Mr Johnson said he held "extensive" discussions on Sunday with Mr Cummings, who he said "followed the instincts of every father and every parent - and I do not mark him down for that".
"Looking at the very severe childcare difficulties that presented themselves to Dominic Cummings and his family, I think that what they did was totally understandable - there's actually guidance... about what you need to do about the pressures that families face when they have childcare needs.
"He found those needs where they could best be served, best be delivered and yes, that did involve travel."
When asked whether Mr Cummings made a trip to Barnard Castle - 30 miles from Durham - during his isolation in April, Mr Johnson said his aide isolated for 14 days and he was "content that in all periods and in both sides (of isolation) he behaved responsibly and correctly".
It strikes me Boris Johnson is taking a political gamble here; that the public will understand his decision or aren't that bothered by a "Westminster row".
Indeed, Mr Johnson and Mr Cummings are seen as political operators who can judge the public mood well.
But many - including several Tory MPs - think they've got this wrong. They believe the public does care and see it as one rule for us, one rule for them.
There are a lot of unanswered questions too; when did the PM know his adviser had travelled to Durham? Did Mr Cummings visit an area 30 miles from where he was isolating?
I suspect this row will continue.
Labour had called for an urgent inquiry into the allegations, while several Conservative backbench MPs publicly questioned Mr Cummings' position, including Sir Roger Gale.
Reacting to the prime minister's comments, Sir Roger said it was an "extraordinary position" for Mr Johnson to take.
"It's up to the prime minister to exercise judgement about who he has around him," he told BBC Radio 5 Live.
"In this case, I do think that that judgement is flawed. I don't think many people will buy into the idea that suddenly after the event it's OK to reinterpret the rules".
'Astonished' with PM
Speaking to the BBC, Labour leader Sir Keir Starmer said: "This was a huge test of the prime minister and he has just failed that test.
"Millions of people across the country have made the most agonising choices - not visiting relatives, not going to funerals - they deserve better answers than they got from the prime minister today."
He also said he would've sacked Mr Cummings if he were prime minister.
Acting Liberal Democrat leader Ed Davey said he was "astonished" with Mr Johnson's decision as the PM had told the public to stay at home.
In a statement posted on Twitter, Amanda Hopgood, leader of the Liberal Democrats on Durham County Council, said "a number of local residents have reported seeing Dominic Cummings on several occasions in April and May".
She said that "given the clear public interest" she has referred the matter to Durham Constabulary to see if there had been a breach of the coronavirus regulations.
And the SNP's Westminster leader Ian Blackford said: "The prime minister's refusal to act demeans his office and will cause lasting damage to public confidence in the Tory government and its response to Covid-19."
Scientists also raised concerns. Stephen Reicher, a professor of social psychology who has advised the government on behavioural science during the pandemic, said the prime minister's comments made him feel "dismay".
He said trust was vital to maintaining public health measures. "You can't have trust if people have a sense of them and us, that there's one rule for them and another rule for us," he told the BBC.
And two Church of England bishops strongly criticised the defence of Mr Cummings. The Bishop of Leeds, the Right Reverend Nick Baines, said the public were being "lied to, patronised and treated by a PM as mugs".
The Bishop of Bristol, the Right Reverend Vivienne Faull, accused the prime minister of having "no respect for people".
Dominic Cummings allegations: A timeline
16 March - Government tells the UK public they have to isolate for 14 days if someone in their household has symptoms
23 March - Boris Johnson tells the UK public they "must stay at home"
27 March - Dominic Cummings seen leaving No 10
30 March - Downing Street says Mr Cummings is self-isolating with coronavirus symptoms
31 March - Officers from Durham Constabulary "were made aware of reports that an individual had travelled from London to Durham and was present at an address in the city", the force adds that officers "made contact with the owners of that address"
12 April - According to the Observer and Sunday Mirror, Mr Cummings was seen visiting Barnard Castle, 30 miles from his parents' residence.
14 April - Mr Cummings is photographed at Downing Street for the first time since 27 March
19 April - This is the date an unnamed witness tells the Observer and Sunday Mirror they saw Mr Cummings in Durham
Earlier, Scottish First Minister Nicola Sturgeon tweeted that the "integrity" of public health advice "must come first" as she urged Mr Cummings to resign.
She added that it was "tough to lose a trusted adviser at the height of crisis", referring to Scotland's chief medical officer who resigned in April after twice breaking lockdown restrictions to drive to her second home.
Some government ministers had rallied around Mr Cummings on Saturday and defended his conduct.
Matt Hancock and Michael Gove were among those to come out in support of Mr Cummings on social media.