David Davis said his resignation as Brexit Secretary will "strengthen" not weaken the Government's negotiating position over Brexit, and quit because he could not back and deliver the Government's plan as he does not believe in.
While Mr Davis' resignation has left Theresa May facing a potential leadership crisis, Mr Davis said he fully supports her as she is a "good Prime Minister".
Despite signing up to the strategy agreed at Chequers on Friday, Mr Davis quit late on Sunday night.
The 69-year-old said he made it "very clear" during the Cabinet meeting that he was the "odd one out" and did not agree with the Prime Minister's plan because there was a "risk of it delivering a poor outcome".
Speaking on BBC Radio 4's Today programme, Mr Davis said that prior to the Buckinghamshire meeting he had let Mrs May know that he was "not at all happy" with her plan, but following a "forthright debate" at Chequers the Prime Minister "won".
He continued that his resignation was one of "principle", and since he did not agree with the Brexit strategy which "gave too much away" to the EU "too easily and that is dangerous", he felt he could not deliver it adequately.
"My fear is that they'll take what we've offered already and then demand more," Mr Davis said.
He added this would leave the Government in a weak position in terms of negotiations.
"I would have been the one who had to deliver this and I would have had to do something I didn't believe in and didn't believe would work," Mr Davis continued.
"The best person to do this is someone who really believes in it, and that is not me."
Rather than leaving the Government in a weaker position in Brexit negotiations - since a new Brexit Secretary will have to be appointed - Mr Davis said his resignation had in fact "strengthened" Mrs May's administration's Brexit position, since someone who "believes" in the plan will now be able to back it and deliver it.
He added that "hopefully a side effect of my departure will be" stopping the Government "from making any more concessions" to the EU over Brexit.
The departure comes as the Prime Minister sets out to win over Tory Brexiteers after a backlash against her plans for leaving the European Union.
Mr Davis' junior ministerial colleague Steve Baker has also quit his role.
In his resignation letter, Mr Davis told Mrs May that he believed "the current trend of policy and tactics" is making it look "less and less likely' that the UK will leave the customs union and single market, adding: "The general direction of policy will leave us in at best a weak negotiating position, and possibly an inescapable one."
The responsibility for leading the negotiations should now go to an "enthusiastic believer in your approach, and not merely a reluctant conscript", he said.
In response, the Prime Minister told Mr Davis: "I do not agree with your characterisation of the policy we agreed at Cabinet on Friday."
She said: "I am sorry that you have chosen to leave the Government when we have already made so much progress towards delivering a smooth and successful Brexit and when we are only eight months from the date set in law when the United Kingdom will leave the European Union."
ITV News Political Editor Robert Peston says Mr Davis' resignation represents the greatest test to Theresa May's authority since she became Prime Minister and that "there may well be a leadership contest before the summer".
Paul Brand, ITV News' Political Correspondent agreed with Peston, tweeting that Mr Davis' resignation is the "biggest challenge" that Mrs May's has faced since the General Election.
While his resignation is thought to have weakened Mrs May's position, Mr Davis told Today that "of course she can survive this".
He added: "I like Theresa May, I think she's a good Prime Minister, but we just have a difference on this... I won't be encouraging people" to pressure Mrs May to resign, "I support her".
Should Mrs May resign, or be forced out as Prime Minister, Mr Davis said he "would not throw my hat into the ring" to replace her.
He added that both he and Mrs May " want to achieve the best for Britain... we just have different ways" of going about this.
Mr Davis' 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 UK closely tied to Brussels.
Arch-Brexiteer Jacob Rees-Mogg, leader of the powerful European Research Group faction within the Tory ranks, said Mr Davis' resignation should force Mrs May to reconsider her approach to Brexit.
He told BBC Radio 5 Live: "These proposals will have to come to the House of Commons in legislation and the question is 'will they command support from Conservative MPs?'
"And I think without David Davis there, without his imprimatur, it will be very difficult for them to get the support of Conservative MPs and therefore the Prime Minister would be well advised to reconsider them."
Fellow Tory Brexiteer Peter Bone welcomed Mr Davis's resignation, saying it was "a principled and brave decision", while Andrea Jenkyns said Mr Davis's departure was "fantastic news" and hailed Mr Baker as "another courageous and principled MP".
Labour leader Jeremy Corbyn said Mrs May "has no authority left and is incapable of delivering Brexit".
Mr Davis was appointed Brexit Secretary by Theresa May in 2016.
On Monday, Mrs May is expected to tell MPs that the strategy agreed on at Chequers is the “right Brexit” for Britain.
But resistance to the plan from hardline Eurosceptics grew over the weekend.
Arch-Brexiteer Jacob Rees-Mogg hit out at the “defeatism” in the Government’s plans, warning that he would vote against them – and suggested other Eurosceptics may do the same.
In the Commons, Mrs May will acknowledge that there have been “robust views” around the Cabinet table and a “spirited national debate” since the 2016 referendum decision to leave the EU.
She will say: “Over that time, I have listened to every possible idea and every possible version of Brexit. This is the right Brexit.”
She will tell MPs it was “the Brexit that is in our national interest” and “will deliver on the democratic decision of the British people”.
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.
But Mr Davis’ decision could now act as a spur to other Brexiteers to take action against Mrs May.
Letters calling for a leadership contest following the Brexit plan have been submitted to the 1922 Committee of Conservative backbenchers, ITV News understands.
Tory MP Andrew Bridgen said: “I can’t support the offer which emerged at Chequers – I think it’s a breach of the red lines, in fact the offer is so poor that I couldn’t support it even if the EU were paying us for it.
“Obviously if the Government and the Prime Minister continue to support that very poor offer then I won’t have any confidence in the Government or the Prime Minister.”
Writing in The Daily Telegraph, Mr Rees-Mogg said: “If the proposals are as they currently appear, I will vote against them and others may well do the same.”
The plan was “the ultimate statement of managing decline” and “focuses on avoiding risk, not on the world of opportunity outside the EU”.
“Pragmatism has come to mean defeatism,” he said.
He also told LBC radio that "if the Government wants to get Chequers through, it will do so on the back of Labour votes, which would be a great mistake.
"There is a large number of Conservative MPs who would not vote for Chequers, because they have committed to their constituents to vote for Brexit."
He added that he had not submitted a letter of no confidence to 1922 Committee chairman Sir Graham Brady.