The Scottish Conservative leader said any regulatory alignment between Northern Ireland and the Republic to keep the border open had to be done on a UK-wide basis.
Ms Davidson warned that "no government of the Conservative and Unionist Party should countenance any deal that compromises the political, economic or constitutional integrity of the United Kingdom".
It was reported that she spoke to the Prime Minister by telephone this morning to relay her concerns.
Ms Davidson said the country should not be divided by "different deals for different home nations" in the wake of the UK Government's attempt to settle the Irish border issue.
The comments came as Nicola Sturgeon urged Labour leader Jeremy Corbyn to join cross-party calls for a 'soft Brexit' that keeps the whole UK inside the single market.
Crunch talks between Mrs May and EU Commission President Jean-Claude Juncker ended without agreement yesterday after the DUP - which props up the minority Conservative Government in Westminster - made clear it would not accept any arrangement which saw Northern Ireland treated differently from the rest of the UK.
It is understood the UK Government and the EU had been poised to agree to guaranteeing ''regulatory alignment'' between Northern Ireland and the Republic in order to prevent the imposition of a hard border.
Leaders of devolved administrations - including Ms Sturgeon - made it clear any special status for Northern Ireland would prompt demands from other parts of the UK for their own tailor-made Brexit.
Ms Davidson said: "The question on the Brexit ballot paper asked voters whether the UK should stay or leave the European Union - it did not ask if the country should be divided by different deals for different home nations.
"While I recognise the complexity of the current negotiations, no government of the Conservative and Unionist Party should countenance any deal that compromises the political, economic or constitutional integrity of the United Kingdom.
"All sides agree there should be no return to the borders of the past between Northern Ireland and the Republic of Ireland.
"Similarly, jeopardising the UK's own internal market is in no-one's interest.
"If regulatory alignment in a number of specific areas is the requirement for a frictionless border, then the Prime Minister should conclude this must be on a UK-wide basis."
Ms Sturgeon said that if one part of the UK can retain regulatory alignment with the EU and effectively stay in the single market, there is "surely no good practical reason why others can't".
She said there was now an opportunity to push for a Brexit deal that could keep the whole of the UK in the single market and customs union, and challenged others to back such a position.
She tweeted: "This could be the moment for opposition and soft Brexit/remain Tories to force a different, less damaging approach - keep the UK in the single market and customs union. But it needs Labour to get its act together. How about it @jeremycorbyn?"