Johnson, who in 2019 won his party's biggest majority in more than 30 years, is now fighting to shore up his authority after a series of revelations about parties in his Downing Street residence during COVID lockdowns.
Johnson has repeatedly apologised for the parties and said he was unaware of many of them.
However, he attended what he said he thought was a work event on May 20, 2020 to which staff had been told to "bring their own booze". Johnson said on Tuesday that nobody had told him the gathering was against COVID rules.
"I expect my leaders to shoulder the responsibility for the actions they take," Brexit-supporting Conservative lawmaker David Davis told parliament.
Davis cited a quote from a Conservative lawmaker, Leo Amery, to then Prime Minister Neville Chamberlain over his handling of war in 1940: "You have sat there too long for the good you have done. In the name of God, go."
Asked directly by an opposition lawmaker if he would resign, Johnson said: "No".
Toppling Johnson would leave the United Kingdom in limbo for months just as the West deals with the Ukraine crisis and the world's fifth largest economy grapples with the inflationary wave triggered by the COVID pandemic, with UK inflation rising to the highest level in nearly 30 years.
To trigger a leadership challenge, 54 of the 360 Conservative MPs in parliament must write letters of no confidence to the chairman of the party's 1922 Committee.
Implored to stay by one supporter, Johnson said he had not yet "sat here quite long enough, indeed nothing like long enough."
As many as 20 Conservative lawmakers who won their seats at the last national election in 2019 plan to submit letters of no confidence in Johnson, the Telegraph reported. A handful of others have already said they had written such letters.
An analysis by The Times newspaper showed that 58 Conservative lawmakers had openly criticised the prime minister.
Leading rivals within the Conservative Party include Chancellor of the Exchequer Rishi Sunak, 41, and Foreign Secretary Liz Truss, 46.
'PORK PIE REVOLT'
Downing Street lockdown parties - some held when ordinary people could not bid farewell in person to dying relatives - have undermined Johnson's authority.
Johnson on Tuesday denied an accusation by his former adviser that he had lied to parliament about a lockdown party, saying nobody had warned him the "bring your own booze" gathering might contravene COVID-19 rules.
"As he waded through the empty bottles and platters of sandwiches - he didn't realise it was a party? Does the prime minister realise how ridiculous that sounds?" Keir Starmer, leader of the opposition Labour Party, told parliament.
"Every week, the prime minister offers absurd and frankly unbelievable defences to the Downing Street parties, and each week it unravels."
Starmer, who welcomed the defection of lawmaker Christian Wakeford who earlier on Wednesday left Johnson's Conservatives to join Labour, asked Johnson if a prime minister should resign if he had misled parliament.
"My decision is about much more than your leadership and the disgraceful way you have conducted yourself in recent weeks," Wakeford said.
"I can no longer support a government that has shown itself consistently out of touch with the hard working people of Bury South and the country as a whole."
Support for Johnson and his party has plummeted after a series of revelations about the parties and other mis-steps.
His own former spokeswoman resigned after she was captured laughing and joking on camera about how to cast a Christmas party if asked about it by reporters.
Such was the revelry in Downing Street at one event that staff went to a supermarket to buy a suitcase of alcohol, spilled wine on carpets, and broke a swing used by the prime minister's young son.
The Mirror said staff had bought a wine fridge for Friday gatherings, events that were regularly observed by Johnson as he walked to his apartment in the building.
Johnson has given a variety of explanations of the parties, ranging from denials that any rules were broken to expressing understanding for the public anger at apparent hypocrisy at the heart of the British state.
He said people must wait for the outcome of an internal investigation by a senior civil servant before reaching conclusions.
Opponents have portrayed him as a charlatan who demanded the British people follow some of the most onerous rules in peacetime history while his staff partied.
The growing internal Conservative rebellion was cast as the "pork pie plot" because one alleged rebel lawmaker was from Melton, the home of the Melton Mowbray pork pie. Pork pie is also London slang for a lie.
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