"It's not practical, it's not desirable and it's not appropriate", Davis said.
They are rallying around a move to give the House of Commons power to send the government back to the negotiating table with Brussels if lawmakers don't like the terms of the Brexit deal struck with the EU.
The vote came on the first of two days of high-stakes debate and votes in the House of Commons on the government's flagship Brexit bill.
On Tuesday, parliament will debate a demand for a "meaningful vote" on any agreement May negotiates with Brussels before leaving the bloc next March.
The victory - by 324 votes to 298 - only came after public haggling between ministers and would-be rebels and a meeting between Mrs May and more than a dozen Tory MPs.
Former deputy prime minister Damian Green wrote in the Sunday Express that the majority of the crucial votes would "pass easily". In my view, this raises the important principle of legitimacy: I do not believe it would be right for the Government to pursue such a course without a plan to seek a confirmatory mandate for the outcome.
A statement from the Department for Exiting the European Union on Tuesday said David Davis had set three tests for any new amendment: not undermining the negotiations; not changing the constitutional role of parliament and government in negotiating global treaties; and respecting the referendum result.
The Labour whips - who enforce discipline in the party - instructed lawmakers Wednesday to abstain on an amendment favoring membership of the EEA.
Scottish National Party (SNP) MPs had earlier walked out of the Commons in protest at what they said was the government's "contempt" for Scotland in the withdrawal process.
Officials from May's Conservative Party have been lobbying lawmakers to reject amendments handed down from the upper house of parliament on her Brexit blueprint, the European Union withdrawal bill, saying they would undermine Britain's negotiating stance.
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"If the Lords amendments are allowed to stand, that negotiating position will be undermined", she told them, adding that she was confident that "I can get a deal that allows us to strike our own trade deals while having a border with the European Union which is as frictionless as possible".
Brexit supporters will be watching closely to see what concessions the government finally offers to the rebels.
Mr Grieve said no government would survive if it tried to dispense with Parliament's input.
Conservative Brexit campaigners accused those in the party who indicated they would vote against the government of not respecting the referendum result.
"On the meaningful vote we have agreed to look for a compromise when this goes back to the Lords", a spokesman said.
In a day of drama, May's position seemed suddenly weaker when junior justice minister Phillip Lee, who has always been critical of the government's Brexit strategy, resigned and said he would vote against the government.
The cumulative effect of 14 Lords amendments which the Government is seeking to overturn could be to "make it impossible to deliver the smooth and orderly exit we want", he warned.
An agreement that defused a potential rebellion over handing parliament more control over Britain's exit from the European Union looked in danger of unravelling on Wednesday, when the two camps argued over the shape of a possible compromise on a "meaningful vote". After losing her party's majority in parliament at an ill-judged election previous year, she now relies on the support of a small Northern Irish party and the distance between victory and defeat is narrow.
May, who leads a minority government propped up by the small Northern Irish Democratic Unionist Party (DUP), conceded that "we need parliamentary support" to implement Brexit.
The most contentious was the bid to give Parliament the power to direct proceedings if the Brexit deal was voted down or no agreement was reached.