Departing Brexit Secretary David Davis disagreed with "the language on Northern Ireland" contained in the EU and UK government's Joint Report in December.
The text which pledged there would be no hard border in Ireland was heralded at the time as a breakthrough in the Brexit negotiations.
Mr Davis resigned late on Sunday night, warning that Britain was giving "too much away, too easily" to the EU.
The outgoing minister said Theresa May's government had gone further than it should have in the negotiations, adopting a "dangerous strategy".
Mr Davis was swiftly followed out of the Department for Exiting the EU by ally Steve Baker as staunch Eurosceptic Dominic Raab was appointed Mr Davis's successor .
In his resignation letter, the former Brexit secretary said he had disagreed with Number 10 on a "significant number of occasions" over the past year or more.
He said he had been at variance with the Tory leader on accepting the European Commission's sequencing of the Brexit negotiations "through to the language on Northern Ireland in the December Joint Report".
Mr Davis said he had "accepted collective responsibility" because part of his role was to "find workable compromises".
He said he also toed the line because he considered it still possible to "deliver on the mandate of the referendum, and on our manifesto commitment to leave the customs union and the single market".
"I am afraid that I think the current trend of policy and tactics is making that look less and less likely," he said.
The former Brexit secretary said the latest UK government white paper would make the "supposed control by parliament illusory rather than real".
"I am also unpersuaded that our negotiating approach will not just lead to further demands for concessions," he said.
Mr Davis yesterday said a leadership challenge would be the "wrong thing to do" and insisted he believed Mrs May was a "good prime minister".
Asked on BBC Radio 4 if she could survive, he replied: "Oh yes, of course."
Downing Street insiders insisted Mr Davis had resigned over a difference of opinion rather than as part of a push against Mrs May, and said he had "done the honourable thing".
Mr Davis said he told Mrs May at Chequers that he was going to be the "odd man out in this" as the latest stage of the Brexit strategy was agreed.
"Hopefully we will resist very strongly any attempt to get any further concessions from us on this, because I think this goes further than we should have gone already," he said yesterday.
Describing the latest plans as having a "number of weaknesses", he explained that it would "not have been plausible" for him to be "front and centre in delivering" it.
