Bill returns to the Commons on Tuesday with ministers seeking to overturn amendments by the House of Lords meant to keep Britain close to the European Union after Brexit.
"I am incredibly sad to have had to announce my resignation as a minister in Her Majesty's government so that I can better speak up for my constituents and country over how Brexit is now being delivered", Mr Lee said on Twitter. May pleaded with MPs, asking them to imagine how weak the government will appear to Brussels bureaucrats if they are defeated on this key piece of Brexit legislation.
The minister also showed a united front with the Prime Minister by denying reports he came close to resigning.
Labour and Conservative MPs are bitterly split over Prime Minister Theresa May's desire to keep Britain in the European Economic Area.More news: Wolfenstein: Youngblood passes the Nazi-killing baton to BJ's twin daughters
"But if the Lords amendments are allowed to stand, that negotiating position will be undermined".
MPs will spend a total of 12 hours debating and voting on 14 Lords amendments to the EU Withdrawal Bill - six hours on Tuesday and six hours on Wednesday.
May, who lost her party's majority at an ill-judged election a year ago, made a last-ditch appeal to lawmakers on Monday and Davis sent a letter making the same case - vote against the government and risk tying Britain's hands in the Brexit talks.
"Whatever we do, we're not going to reverse that", Davis told BBC radio. The government said it would not support that amendment.More news: Dwane Casey to Coach Pistons
The MP says he will back a knife edge vote created to give parliament the power to vote down May's Brexit deal.
"We are asking members of parliament to abide by the referendum result, our manifesto commitment and to back our country", Andrew Bridgen, Conservative lawmaker and Brexit campaigner, told Reuters. She had nearly lost a general election she never needed to call, squandering her Conservative Party's overall majority and forcing her to rely on the support of the smaller Democratic Unionist Party.
It is expected that Tuesday will see MPs decide whether Parliament should have the power to set the Government's negotiating goals if Theresa May's deal with Brussels is voted down.More news: Donald Trump: Top economies should drop all tariffs