Mr Johnson accused Sir Keir of being in a "Doctor Who time warp" as the Labour leader repeatedly probed him on why he has not resigned over Partygate proceedings.
It came as Sir Keir opened Prime Minister's Questions (PMQs) by asking about the resignation of Mr Johnson's former spokeswoman Allegra Stratton.
He asked: “Why did the Prime Minister’s press secretary Allegra Stratton have to resign from her job?”
The PM replied: “I bitterly regret Allegra’s resignation.
“I think it was very sad and I think she did an outstanding job… particularly since she was the one who coined the expression ‘coal, cars, cash and trees’ which enabled the UK to deliver a fantastic COP26 summit last year.”
Sir Keir continued to push the PM on why he hasn't offered his resignation over Partygate, to chants of "beer" from the Tory benches – in reference to a photograph of the Labour leader with a pint of beer during the Covid lockdown.
Boris Johnson in the Commons today POOL
Sir Keir noted that Professor Neil Ferguson resigned, as did Ms Stratton, but neither Chancellor Rishi Sunak nor Mr Johnson have followed suit.
Sir Keir asked: “Does the Prime Minister actually accept that he broke the law?”
Mr Johnson hit back though, comparing Sir Keir to Labour predecessor Jeremy Corbyn after making the Doctor Who quip.
The PM said: “I think he is in some kind of Doctor Who time warp.
"We had this conversation yesterday, Mr Speaker, and I have explained why I bitterly regret receiving an FPN (fixed penalty notice) and I apologised to the House.”
He added he would “get on with delivering for the British people” and “power out of the problems that Covid has left us”.
The PM then faced a grilling over reports that he had criticised the BBC over their coverage of the war in Ukraine.
Sir Keir accused Mr Johnson of opting to “slander decent people” in private but lacking the “backbone to repeat it in public”.
He told the Commons: “He never takes responsibility for his words or actions. They were all there. The Prime Minister also accused the BBC of not being critical enough of Putin.
“Would the Prime Minister have the guts to say that to the face of (BBC reporters) Clive Myrie, Lyse Doucet and Steve Rosenberg, who have all risked their lives day in, day out on the frontline in Russia and Ukraine uncovering Putin’s barbarism?”
Mr Johnson replied: “If (Sir Keir) wants to join the Conservative Party and come and listen to the meetings of the Conservative Party he’s welcome to do it though, as I say, I think he’s a Corbynista in an Islington suit.
Keir Starmer in the Commons today POOL
Sir Keir repeatedly called on the PM to resign House of Commons
Boris Johnson erupted at Sir Keir's swipes House of Commons
“But I said nothing of the kind and I have the highest admiration as a former journalist for what journalists do. I think they do an outstanding job. I think he should withdraw what he just said – it has absolutely no basis or foundation in truth.”
Sir Keir countered: “That’s how he operates: a mealy mouthed apology when the cameras roll, a viscous attack on those who tell the truth as soon as the cameras are off. Slander decent people in a private room, let the slander spread without the backbone to repeat it in public.”
Later in the session, SNP Westminster leader Ian Blackford asked Mr Johnson if the Scottish public are right to believe he lied about breaking Covid laws.
Mr Blackford said: “Last night the Prime Minister might have convinced his backbenchers and his spineless Scottish Tories to keep him in place for another few weeks, but the public aren’t so easily fooled.
“82 percent of people in Scotland said they believe the Prime Minister lied to this Parliament and to the public about his lawbreaking Covid parties. Are they right? Or should they not believe their lying eyes?”
The PM replied: “We had a long conversation about this yesterday. I understand the point of his question, but we are going to get on with the job of delivering for the people of the whole of the United Kingdom.”
During proceedings, every party leader also took the time to wish the Queen a happy birthday as she turns 96 tomorrow.
It comes as Labour is poised to put forward its motion seeking to refer Mr Johnson for investigation by the Privileges Committee to determine whether he misled Parliament.
Sir Keir will hope to capitalise on the political leverage this offers, as the Committee has wide-ranging powers and could ask to look at the Sue Gray report.
It also has the power to recommend sanctions.
Mr Johnson appeared in the Commons yesterday, where he said sorry after the Met Police fines.
But Sir Keir called the apology a “joke” and Tory MP Mark Harper broke ranks from his colleagues by calling on the PM to resign over his “indefensible” actions.
The Labour leader went on to bring up the story of John Robinson, whose wife died of Covid.
Sir Keir said John "followed the Prime Minister's rules" and didn't visit his wife in hospital in an address that visibly moved the Tory benches.
But speaking to GB News' Tom Harwood, Tory MP Michael Fabricant hit out at Sir Keir and other Labour MPs for "choosing to weaponise the personal tragedies endured by people".