His resignation comes just two days after Mrs May secured the backing of her cabinet - including Mr Davis - for a plan to keep close ties to the European Union after leaving the bloc.
The two departures seemed to shatter May's own proclamation of cabinet unity last Friday, when she said she believed she had, after two years of wrangling, secured agreement on Britain's biggest foreign and trading policy shift in nearly half a century.
"As far as negotiating the deal, I probably would have done what my suggestion was to the prime minister, but she can always do that".
The EU's chief negotiator, Michel Barnier, has insisted the proposals must be workable.
Yet after reflecting on the situation, Mr Davis and Mr Baker apparently decided they could not live with such a soft Brexit after all.
Trump's interview easily could overshadow the government's attempt to lay out plans for what it calls a "principled and pragmatic" Brexit. "That dream is dying, suffocated by needless self-doubt".
Britain "is in somewhat turmoil", Trump said before departing Washington, remarking that dealing with Putin might surprisingly be the easiest part of the European trip. "In that respect we are truly headed for the status of colony".
The pound was up 0.5% against the dollar at 1.33, and 0.3% higher versus the euro at 1.13.
Above a picture of her cabinet, including her new foreign secretary, Jeremy Hunt, and Brexit minister Dominic Raab, May tweeted: "Productive Cabinet meeting this morning - looking ahead to a busy week".
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.
The White Paper, reportedly delayed for several months amid disagreements between ministers, was finally approved by Theresa May's cabinet at her country residence, Chequers, last week.
During a session in parliament, May signalled that, after finally nailing her colours to a vision for Brexit, she had chose to face down the dissenters, who do not form a majority in parliament.
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Asked if the prime minister could survive.
"Over that time, I've listened to every possible idea and every possible version of Brexit".
Trump said that the first thing he had done upon his arrival at Chequers on Friday was to offer a mea culpa to May but that she had assured him none was necessary, joining him in pinning the drama on the news media.
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.
"We achieved all of the things that we campaigned for in order to ensure we could leave the European Union, but we also do so in a way which respects some of the wishes and some of the concerns of some of my colleagues that voted remain".
He said that felt he had to resign as he had no desire to be a "reluctant conscript" in the selling the PM's deal in negotiations with the EU.
"But I come back to this point, if people don't like this proposal what is their alternative?"
The UK is set to leave the European Union on 29 March 2019, after the 2016 referendum in which people voted by 51.9% to 48.1% for Brexit. On Tuesday, two more lawmakers followed them out the door.
They fear a clean break would cost jobs.
"There have been differences within the Conservative Party over how far the United Kingdom should prioritise the economy by compromising on issues such as leaving the remit of the European Court of Justice and ending free movement of people".
"This is not a betrayal".