The Prime Minister joined the Labour leader in the House of Commons as a raucous atmosphere took hold in the Chamber.
Responding to a chorus of cheers from the opposition benches as he opened proceedings, Sir Keir said: “Awful behaviour, unacceptable in any walk of life, it's there for all to see but he ignores it.
“It was the same when his ally was on the tape from the lobbyists.
“It was the same when the Home Secretary was bullying staff, it was the same when taxpayers money was being abused.
“And it was the same when he and his mates were partying during lockdown.
Boris Johnson and Sir Keir Starmer GB News
“Anyone quitting now after defending all that hasn’t got a shred of integrity.
“Isn’t this the first recorded case of the sinking ships fleeing the rat?”
Mr Johnson had to contend with a luke-warm reception from his own party, but still enjoyed enthusiastic cheers from loyalist Tories choosing to back the PM.
He fired back at the Labour leader: "He should hear about what his lot say about him.
"He talks about integrity, he wanted to install the member for Islington North (Jeremy Corbyn) into No10, imagine what our country and the world will be like now.
"He talks about integrity, he voted 48 times to overturn the will of the British people and take us back into the European Union and listening to his muddled speech the other day that is exactly what he would do again.
"He talks about integrity but he has voted times and times against sanctions against criminals that will put them behind bars.
Sir Keir described resigning Tory ministers as sinking ships fleeing the rat House of Commons
"He is himself facing a criminal investigation for which he asked me to resign."
It comes after a turbulent 24 hours for the Government and the embattled Conservative Party leader.
A string of high-profile departures rocked the country on Tuesday evening, led by ex-Chancellor Rishi Sunak and former Health Secretary Sajid Javid – who delivered broadsides at Mr Johnson as they quit their Cabinet posts.
The Prime Minister's handling of the row over scandal-hit former Deputy Chief Whip Chris Pincher became the latest issue to raise questions over his judgment.
During the debate in the Commons, Mr Johnson said he is not going to “trivialise what happened” when asked if he ever said “Pincher by name, Pincher by nature” by Sir Keir.
The Labour leader asked him: “None of that explains why he promoted him in the first place. And we have heard it all before. We know who he really is. Before he was found out, he has reported to have said he is handsy. That’s the problem. Pincher by name, Pincher by nature.
“Now, has the Prime Minister ever said words to that effect? And I’m not asking for bluster and half-truth. We’ve all had enough of that. Yes or no?”
Rishi Sunak and Sajid Javid both resigned from ministerial roles this week Toby Melville
The Prime Minister replied: “I am not going to trivialise what happened. Yes, because the very serious complaints have been raised against the member for Tamworth, and they’re now being investigated. It is true. It is true that the complaint was raised when he was in the Foreign Office and the matter was resolved. It is absolutely true.
“It’s absolutely true that it was raised with me. I greatly regret that he continued in office, and I have said that. I have said that before. I have said that before, but it is now a subject of an independent investigation, and that is the right thing.”
While Mr Johnson told MPs his job is to “keep going” after a Conservative backbencher asked in what circumstances he would resign.
Conservative former minister Tim Loughton asked: “Does the Prime Minister think there are any circumstances in which he should resign?”
The Prime Minister replied: “Clearly if there were circumstances in which I felt it was impossible for the Government to go on and discharge the mandate we have been given, or if I felt, for instance, we are being frustrated in our desire to support the Ukrainian people, or over some related point, then I would.
“But, frankly Mr Speaker, the job of a Prime Minister in difficult circumstances when he has been handed a colossal mandate is to keep going and that is what I am going to do.”
In his resignation letter, Mr Sunak said “the public rightly expect government to be conducted properly, competently and seriously”.
He added: “I believe these standards are worth fighting for and that is why I am resigning.”
Nadhim Zahawi was promoted to be the new Chancellor, with universities minister Michelle Donelan taking his place as Education Secretary.
Mr Javid said the British people “expect integrity from their government” but voters now believed Mr Johnson’s administration was neither competent nor “acting in the national interest”.
He was replaced as Health Secretary by Steve Barclay, the Prime Minister’s chief of staff.
On Wednesday morning, more resignations came before the Parliamentary session began.
Will Quince resigned as children and families minister, saying he could not accept being sent out to defend the Prime Minister on television with inaccurate information about Mr Pincher.
While Laura Trott quit as a ministerial aide, saying “trust in politics is – and must always be – of the utmost importance, but sadly in recent months this has been lost”.
Schools minister Robin Walker, Treasury minister John Glen and Minister for Prisons and Probation Victoria Atkins also stepped down earlier today.