However, he also reportedly threatened to resign from government on a number of occasions, most recently during talks to resolve a dispute with pro-EU Conservative lawmakers who wanted a greater say for parliament over the final Brexit deal.
But Mr Davis, who signed up to the plan agreed by the cabinet at Chequers on Friday, has now quit. There was no immediate comment from Baker.
May is due to address parliament later on Monday to explain her plan for Britain to adopt European Union rules on goods after Brexit, amid anger from MPs in her own party who want a cleaner break and businesses who say it may still cause economic harm.
In a resignation letter circulated by local reporters, Davis told May he has repeatedly disagreed with the government over the past year and that it is looking "less and less likely" that the government can deliver Brexit per voters' wishes. In 2008, when the Conservatives were not in government, Davis quit as a member of parliament to raise the profile of a debate over what saw as the erosion of civil liberties.
With nine months before Britain leaves and just over three before the European Union says it wants a deal, May has been under intense pressure from the bloc and from many businesses to show her negotiating position.
May is due to brief lawmakers Monday on the plan hammered out during a 12-hour meeting at Chequers, the prime minister's country retreat.
May's plan would create a free trade area with the European Union for goods, to protect supply chains in areas such as manufacturing, while maintaining flexibility for Britain's dominant service sector.
Brexit-backing lawmakers have been angered by May's plans, saying they will keep Britain too close to the European Union and limit its ability to strike new trade deals.
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.
Conservative MP Jacob Rees-Mogg, a leader of the party's "hard Brexit" faction, compared May's plan to an egg so softly boiled that it "isn't boiled at all".
"A very soft Brexit means that we haven't left, we are simply a rule-taker", he said.
Some dream of replacing May with a staunch Brexiteer, such as Foreign Secretary Boris Johnson, who in the past has disagreed publicly with his boss.
The Prime Minister will insist the plan, which would see the UK share a "common rulebook" for goods as part of a proposal to create a UK-EU free trade area, still meets her Brexit red lines. Michael Gove, May's environment minister, said on Sunday that while the agreed negotiating stance was not ideal, he believed it delivered on handing back control to Britain.
Mastermind behind 1995 Japan sarin-gas attack executed
Two days after the attack police raided the compounds of the Aum Shinrikyo cult. "I've been in pain for years", he told AFP. Asahara guided the attack, according to testimony by his right-hand man, Yoshihiro Inoue, who also was executed Friday.
Bill Shine joins White House as communications director
Shine stepped down at Fox News past year after he was criticized for how he handled sexual harassment claims at the network. Reaction to Shine's hiring included some negative responses from those who used to work for him or Fox News .