At a Cabinet meeting Friday, May demanded members support a plan for Brexit that it would include Britain partly remaining in the single market and signing up to a "common rulebook" for goods.
In 2016, following the resignation of Prime Minister David Cameron, Johnson was widely considered a front-runner in the race that May eventually won, but he ruled himself out after the defection of a key ally.
Detailed talks were held between Number 10 and MPs on Saturday in a bid to reassure backbenchers over the Government's plans for Britain's future relationship with the European Union, while further meetings took place on Sunday and Monday lunchtime.
'As I said then, the Government now has a song to sing. This is complete and utter chaos. If a country cannot pass a law to save the lives of female cyclists - when that proposal is supported at every level of UK Government - then I don't see how that country can truly be called independent. "The dream is dying, suffocated by needless self-doubt", he wrote.
Mr Johnson had yet to appear, 20 minutes after news of his resignation was announced.
Liberal Democrats leader Sir Vince Cable said: "He might have taken time to pluck up the courage to join his hard Brexiteer colleagues, but Boris Johnson's resignation completely destroys the Conservative position".
"What we are proposing is challenging to the European Union", she said.
In his letter to the Prime Minister, Davis said: "Whether it is the progressive dilution of what I thought was a firm Chequers agreement In February on right to diverge, or the unnecessary delays of the start of the white paper, or the presentation of a backstop proposal that omitted the strict conditions that I requested and believed that we had agreed, the general direction of policy will leave us in at best a weak negotiating position, and possibly an inescapable one".
"The question is how do you leave and is this going to be a proper Brexit?"
"If what the Lord says is correct he has been a towering figure in the Government". Since I can not in all conscience champion these proposals, I have sadly concluded that I must go.
British Prime Minister Theresa May has named Jeremy Hunt to the job of foreign secretary after the resignation of Boris Johnson.
"It is clear this Government can not secure a good deal for Britain".
"And that is just within her own Cabinet".
Mr Corbyn added: "There are now only a few months left until these negotiations are supposed to conclude".
Theresa May's government is in crisis. It is clear. this Government is not capable of securing a deal to protect the economy, jobs and living standards. Parliament will decide whether or not to back the deal the Government negotiates, but that deal will undoubtedly mean the returning of powers from Brussels to the United Kingdom.
Matt Hancock had been culture secretary since January this year, and had previously been a junior minister in the same department.
Mr Davis said it was a "pity" that the Foreign Secretary had quit. I had resigned because this was central. I'd have to present it in Europe.
But leading pro-Brexit legislator Jacob Rees-Mogg said "I don't think a no-confidence vote is immediately in the offing". "It's a pity, but there we are". May appointed staunchly pro-Brexit lawmaker Dominic Raab to replace Davis, who insisted he did not want his resignation to become a rallying cry for the prime minister's ouster.
Jeremy Hunt visited the Foreign Office last night.