The Environment Secretary is a prominent Brexiteer and - hours before Mr Davis's resignation - on Sunday urged fellow Tories to get behind the Prime Minister and her Brexit plan agreed at a Chequers summit on Friday.
But former Conservative leader William Hague, writing in the Daily Telegraph, said May´s critics had failed to produce "any credible alternative proposal" and warned that further resignations could put Brexit itself "at risk".
His resignation has been followed by that of Brexit ministers Steve Baker and Suella Braverman.
A contest would only be held if she loses a vote of no confidence, triggered by at least 48 MPs sending letters to the chair of the backbench 1922 Committee.
Earlier, May defended her Brexit plan to lawmakers in the House of Commons, with Johnson absent from his usual place on the Conservative front bench.
"This is not a betrayal", she responded to one of several eurosceptic Conservative MPs who complained, insisting it was "the right Brexit deal for Britain".
On Friday, her cabinet ostensibly agreed to pursue a close post-Brexit trading relationship with the EU.
In response to questions about whether the government is in "meltdown", May's spokesman said "it is not". "The trouble is that I have practiced the words over the weekend and find that they stick in the throat".
It comes right before Donald Trump's first visit to Britain as United States president on Friday, as well as the North Atlantic Treaty Organisation summit in Brussels tomorrow.
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All eyes are now on the next move by Brexit hardliners in May's centre-right Conservative Party.
After his bruising stint at the Foreign Office, even some of Johnson's erstwhile backers have drifted away, and most Tory MPs believe he is unlikely to muster a large enough cohort of supporters to fire off sufficient letters. Raab was previously a minister for housing.
May said: "What we are proposing is challenging for the EU".
People's Vote will deliver the signatures to Downing Street a week before the the deal is presented at Parliament in autumn.
Tusk tweeted: "Politicians come and go but the problems they have created for people remain". "But. who knows?", he added, wondering if Britain might reject the divorce.
Johnson quit Monday with an incendiary letter accusing May of killing "the Brexit dream" and flying "white flags" of surrender in negotiations with the European Union.
"In my view the inevitable effect of the proposed policies will be to make the supposed control by Parliament illusory rather than real".
May must now quickly seek European Union support for her proposal.
Davis said his job required "an enthusiastic believer in your approach, and not merely a reluctant conscript".
The Sun, Britain's biggest-selling newspaper, which has long argued for Brexit, said May had made "blunder after blunder". Both fiercely criticised her negotiating stance.
The government has battled to find a way of tackling these while maintaining access to European Union markets and avoiding burdensome customs arrangements that could damage businesses, with some leading companies warning they could cut investment or withdraw from Britain in the wake of a no-deal Brexit.