The Cabinet Secretary is effectively Mrs May's deputy and has previously held the position of Europe Minister in the Foreign Office.
He said on Monday he thought May was a good prime minister and did not want to see her replaced, but could not defend her strategy in good faith.
Davis said he could not support May's plan to maintain close trade and regulatory ties with the European Union, which he said gave "too much away, too easily".
In her reply, Mrs May told him: "I do not agree with your characterisation of the policy we agreed on at Cabinet on Friday".
Fallon warned Conservative rebels that a challenge to May's leadership is "the last thing we need".
"We need to support her and let her deliver what the country voted for, which is to leave the European Union".
He has served as health secretary since 2012.
But the Press Association understands that Mr Davis, who signed up to the plan agreed by the Cabinet at Chequers on Friday, has now quit.
His resignation was warmly welcomed by hardline Eurosceptics in the Tory ranks who were already expressing reservations about Mrs May's leadership after her Cabinet agreed a plan which would keep the United Kingdom closely tied to Brussels.
British Prime Minister Theresa May met with her shaken-up Cabinet on Tuesday as she tried to restore government unity after the resignations of two top ministers over Brexit.
Of course it is still possible that May will ultimately decide of her own volition to quit.
Less than nine months remain until Britain reduces the EU's membership on March 29, 2019.
The "common rulebook" plan "hands control of large swathes of our economy to the European Union and is certainly not returning control of our laws in any real sense", he wrote to Mrs May.
The Prime Minister is expected to appoint a replacement for Mr Davis on Monday.
Raab, who is promoted from housing minister, now faces the hard task of calming nervous Brexiteers within the Tory ranks who are threatening to derail Theresa May's plans.
"It seems to me we're giving too much away, too easily, and that's a unsafe strategy at this time", Davis said in a BBC radio interview Monday morning.
But leading pro-Brexit legislator Jacob Rees-Mogg said "I don't think a no-confidence vote is immediately in the offing".
"It is as though we are sending our vanguard into battle with the white flags fluttering above them", Johnson wrote in a letter that underscored his credentials as a champion of full-speed Brexit.