Ms Davidson has been lauded after helping boost the number of Tory seats to 13 north of the border up from one. By contrast Mrs May is under fire after her election gamble "backfired" with the loss of 13 seats UK-wide wiping out the party's Commons majority.
The Scots Tory leader today ruled out any prospect of running for Westminster and challenging for the UK leadership, insisting she is "100%" behind Mrs May.
But she said that the party must "listen to voters" who did not give the UK Conservative Party the mandate it sought in the election.
Mrs May has previously set out an uncompromising approach to the Brexit talks, indicating "no deal is better than a bad deal."
But Ms Davidson said: "It is incumbent on us to listen to other parties in Parliament, and people outside it, about the best way forward.
"It is the policy of both the Conservative party and the UK Labour party to respect the Brexit referendum result.
"That means we are leaving the European Union. We must leave the Common Fisheries Policy.
"At the same time we must in my view seek to deliver an open Brexit, not a closed one, which puts our country’s economic growth first.
"I am confident that, once the dust settles on last night’s result, we can rise to that challenge. The truth is that we are still the only UK party with a plan for Britain."