Ruth Davidson has said she fears a second EU referendum could set a precedent in Scotland.
The Scottish Tory leader warned failing to abide by the result of the 2016 vote risks the public’s faith in democracy.
Ms Davidson questioned how pro-union politicians could oppose a second Scottish independence referendum if they back another EU ballot.
Her comments come a day after Mrs May announced she would offer MPs a vote on whether or not to hold a second Brexit referendum in a bid to get her Brexit Withdrawal Agreement Bill through parliament.
Writing in the Scottish Daily Mail, Ms Davidson said: “If we really do respect democracy in this country then political parties should try to accept the need to deliver Brexit as promptly as possible.
“The failure to do so has, justifiably, sparked a backlash in Scotland and right across the UK.
“We must listen. I’m utterly clear in my mind that we should abide by the result of the 2016 referendum or risk faith in that democracy withering on the vine.”
She added: “I also fear the precedent that a second EU referendum would set if we do not.
“How can pro-union politicians like me credibly oppose indyref2 if our position is to back another EU referendum?
“I voted No and Remain in these polls. But how can you demand a rerun of a referendum result you lost but insist the one you won is upheld?”
SNP leader Nicola Sturgeon highlighted the apparent division at the top of the Tory party on the issue of a second Brexit referendum.
She tweeted: “The PM might not have guaranteed a second EU referendum, but her promise of a vote on one puts the Scottish Tories ‘vote for us for no more referendums on anything ever’ pitch on something of a sticky wicket. Just an amusing aside.”