In her response, Mrs May said that "I do not agree with your characterisation of the policy we agreed at cabinet on Friday". That plan is now in tatters and her political future appears uncertain.
Labour MP Andrew Adonis said Raab's appointment could spell the "death" of the Tory leader, while writer and editor Hitcham Yezza said Brexit has now become even more "shambolic".
'That dream is dying, suffocated by needless self-doubt'.
British Prime Minister Theresa May talked with British Foreign Minister Boris Johnson, in May 2017, with British lawmaker Michael Fallon, right, as they participated in a North Atlantic Treaty Organisation summit of heads of state and government in Brussels.
Mrs May insists her plans are the "the best way to honour" the Brexit vote, amid a deepening political crisis.
Davis's late-night resignation undermines May's fragile government.
He said: 'We are now in the ludicrous position of asserting we must accept huge amounts of precisely such European Union law, without changing an iota, because it is essential for our economic health - and when we no longer have any ability to influence these laws as they are made. "I'd have to do something I didn't believe in", he told the BBC.
He told the Prime Minister: 'As I said then, the Government now has a song to sing.
British journalist and writer Neil Clark told RT how there is now "incredible uncertainty" over how the United Kingdom will go about Brexit negotiations, given a "split" in the party over the country's future trade relations with the EU.
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Johnson said crucial decisions on preparing for a no deal Brexit had been "postponed" - meaning Britain was headed for a "semi-Brexit".
Rebuffing claims that her proposals make too many concessions to the EU, May said "this is the right Brexit" and would leave Britain free to make its own laws and trade deals.
But as Foreign Secretary in May's government, he was prone to gaffes and criticized for not being on top of his brief.
There was no immediate statement from Johnson, another loud pro-Brexit voice within May's divided government.
"What Theresa May was trying to do with her proposal last week was to come up with a compromise that would keep all sides on ball within the Conservative party, and clearly that has failed", Clark said.
Ministers gathered for a retreat at the premier's country residence, Chequers, on Friday for talks called at short notice, and mobile phones had to be left outside.
Mr Davis's deputy, Steve Baker, also resigned.
Instead May was tonight scrambling to shore up her ailing Premiership amid fierce rumours she is set to face a no confidence vote, possibly within hours.
Under Conservative Party rules, a confidence vote in a leader can be triggered if 48 Conservative lawmakers write a letter requesting one.
May does not enjoy a majority in Parliament, and governs only with the support of the 10 MPs of the Northern Irish Democratic Union Party (DUP).