Phillip Lee said the move was due to his intention to back a plan giving parliament power to direct Ms May’s approach, something the prime minister is determined to avoid.
He also went directly against official policy in saying that the government should “go to the people, once again” for a decision on the final Brexit deal.
The ex-justice minister said he is “incredibly sad” to be leaving his job, but argued that he could “better speak up” for his constituents on EU withdrawal once off the government payroll.
Despite demands for unity from Downing Street, Mr Lee received some immediate support from colleagues on the back benches, with his departure even sparking muttering that other junior ministers may walk.
Mr Lee has openly said that he backed Remain in the last referendum and has previously criticised the government, saying that it should focus on “evidence, not dogma”.
In a series of heartfelt tweets on Tuesday, he wrote: “If, in the future, I am to look my children in the eye and honestly say that I did my best for them I cannot, in all good conscience, support how our country’s current exit from the EU looks set to be delivered.”
He went on: “When MPs vote on the House of Lords’ amendments to the EU Withdrawal Bill, I will support the amendment which will empower parliament to reject a bad deal and direct the government to re-enter discussions. For me, this is about the important principle of parliamentary sovereignty.
“Then, when the government is able to set out an achievable, clearly defined path – one that has been properly considered, whose implications have been foreseen, and that is rooted in reality not dogma – it should go to the people, once again, to seek their confirmation.”
His shock departure came as David Davis warned potential Tory rebels that they cannot undo the EU referendum, ahead of a tricky 48 hours in which the government will try to get its Brexit programme back on track.
The Brexit secretary spoke after some Conservatives signalled that they would support an amendment to the EU (Withdrawal) Bill tabled late on Monday by former attorney general Dominic Grieve giving more power to MPs.
The Commons begins two days of debate and votes on the bill on Tuesday and Mr Grieve’s “meaningful vote” amendment calls for a binding motion to be passed, setting out how to proceed in the event of a “no deal” Brexit, in which case MPs – and not the executive – would call the shots.
But Mr Davis told BBC Radio 4’s Today programme: “A meaningful vote is not the ability to reverse the decision of the referendum.
“We will put in front of parliament the decision for them to vote … after that there will be a process of primary legislation to put the actual details of it in parliament, so parliament will actually decide on the application of the detail.”
Asked what would happen if they voted against the deal, he said: “If they throw it out, well, they throw it out. We will have to go away, think about it and come back and make a statement, which is what I am saying to the house this afternoon.”
The prime minister appeared to have defused a potentially explosive row over the EU customs union on Monday night as Tory pro-Europe rebels Sir Oliver Letwin and Nicky Morgan and Brexiteers Jacob Rees-Mogg and Sir Bill Cash came together to table a separate compromise amendment backing “a customs arrangement” with the EU.
Ms May set the tone for a tense two days as she warned Tory Remainers that if they defied her and backed Lords amendments to the landmark Brexit legislation it would weaken Britain’s hand at the negotiating table.
Addressing a meeting of the backbench 1922 Committee on Monday, ahead of the key votes, she said: “I am confident I can get a deal that allows us to strike our own trade deals while having a border with the EU which is as frictionless as possible.
“But if the Lords amendments are allowed to stand, that negotiating position will be undermined.”