In a statement, May's spokesman said the prime minister will fight any attempt to oust her.
Health Secretary Jeremy Hunt has been named as Mr Johnson's replacement as foreign secretary.
"I'm pleased David Davis has resigned, I'm pleased that Boris has made a decision to step aside and I think this Prime Minister needs to go".
Davis said he could not support May's plan to maintain close trade and regulatory ties with the European Union, which he said gave "too much away, too easily".
May's personal authority was badly damaged after she called a snap election previous year to strengthen her hand in Brexit talks but instead she lost her parliamentary majority, meaning she has to rely on a small Northern Irish party to govern.
In his resignation letter, Johnson harshly criticised the government's approach to Brexit: "Brexit should be about opportunity and hope". He built a public profile on Latin quips and rumpled eccentricity, while nurturing a poorly concealed ambition to lead his country.
The resignations were a stunning reversal from Friday's special Cabinet at Chequers when all ministers signed up to the Brexit plan.
In the June 23, 2016 referendum, 17.4 million voters, or 51.9 percent, backed leaving the European Union, while 16.1 million voters, or 48.1 percent, backed staying.
She looks set on facing down a rebellion in her Conservative Party, where hardline Brexit supporters are livid over her plans to negotiate "a free trade area for goods" with the EU. Northern Ireland is part of the United Kingdom.
"The general direction of policy will leave us in at best a weak negotiating position, and possibly an inescapable one", he said.
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".
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Following her and Harry's engagement announcement last November, it was revealed that Meghan would become a United Kingdom citizen.
Although it has caused a headache, May was quick to capitalize on the fact that two of her biggest problems extricated themselves from her inner circle; she ceremoniously replaced Davis and Johnson with more amiable substitutes.
Less than nine months remain until Britain reduces the EU's membership on March 29, 2019. They also lack the support in parliament to try to change any deal that is agreed with the EU.
There was a crisis in government, he said, adding: "It is clear this government can not secure a good deal for Britain".
"I don't think we should be distracted now by some of the political challenges that are going on at Westminister".
Steve Baker, a junior Brexit minister, also resigned.
Senior lawmakers in her party said they did not expect the prime minister to face a vote of no confidence, although some Conservatives were still saying that she should go.
"In retrospect he should have never have been Foreign Secretary, he had to be in the government given his Brexit campaign, but he should have been in charge of a domestic department".
"I'd have to deliver this". "That's not a tenable position".
But if he decides he wants a tilt at the Tory leadership after bottling it at the last minute in 2016, then Theresa May's fate is likely to rest with a group of ruthless and single-minded backbench MPs - the European Research Group (ERG).
May, the leader of the Conservative party, believed she had secured a hard-won agreement with her deeply divided cabinet of ministers on Friday to keep the closest possible trading ties with the EU.
But she wakes up this morning facing the toughest challenge of her time in Downing Street to date. He urged May to abandon her plans and take a tougher line with Brussels.