The former Foreign Secretary accused the Prime Minister of "suffocating" the Brexit "dream".
"His replacement will be announced shortly".
"If the United Kingdom is able to relax some of its red lines, then the European Union should be flexible too", Irish Prime Minister Leo Varadkar told a session of the Irish parliament.
"It would deliver on both sides' commitments to Northern Ireland and Ireland, avoiding a hard border without compromising the EU's autonomy or the UK's sovereignty".
The departures raise the stakes for May, who believed that 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.
British Prime Minister Theresa May moved to reassert her authority on Tuesday after two top cabinet members quit and launched broadsides against her Brexit plans, winning support from many of her ministers including a leading eurosceptic. "However, even in that event, it seems to me that the national interest requires a secretary of state in my department that is an enthusiastic believer in your approach, and not merely a reluctant conscript".
Pompeo sees hard road ahead but pursues N.Korean denuclearisation talks
America has about 28,500 troops based in South Korea and they routinely train with their local counterparts. However, he did not go into detail about what was actually achieved during his latest stay in Pyongyang.
Under the plans, Britain would stick to a "common rulebook" with the European Union for goods and agricultural products in return for free trade, without tariffs or border customs checks.
This was too close to the European Union for "Hard Brexiteers" such as David Davis - as many in her party have made clear in the days since and the likes of Johnson, who campaigned for Brexit and has been one of its most passionate cheerleaders since. May was forced to reshuffle her cabinet after Boris Johnson put in his papers on Monday.
Andrew Bridgen became the first Conservative MP to declare publicly that he has sent a letter of no confidence in Mrs May to the chairman of the backbench 1922 Committee, Sir Graham Brady.
The Cabinet minister said service agreements almost always allow "some mode for movement that goes with that" but that was a "million miles away" from European Union free movement.
Amid chaotic scenes, the Commons was briefly suspended as Brexit Secretary Dominic Raab tried to present the policy paper before it was given to MPs.
"We will not get a clean, democratic Brexit under Theresa May and if the Conservative Party don't come to their senses, then yes, when there is a leadership contest in Ukip at the end of March next year, then I would seriously consider throwing my hat back in the ring".
Britain voted for Brexit in June 2016, but May has so far been unable to present a common position to Brussels on what she wants because of deep divisions in her government.
Writing in the Daily Telegraph she said Mrs May's plan was "catastrophically bad" and a "disaster for the Conservative Party". Steve Baker also resigned as a junior Brexit minister.