Jackson Carlaw, who has assumed day-to-day leadership duties, said an “introspective” debate on the future of the party at a crucial moment for Brexit would not play well at the ballot box.
He spoke out after the prospect of loosening ties with the UK party – and even changing its name in Scotland – was raised by two senior Scottish Tory MSPs after Ms Davidson quit.
Murdo Fraser and Adam Tomkins, who are both seen as possible contenders for the leadership, both believe the plan should be discussed to allow MSPs to differentiate themselves from the UK party.
Mr Fraser first raised the idea when he stood against Ms Davidson in 2011, arguing for the Tory name to be scrapped and a new right-of-centre party to be launched in its place.
Professor Tomkins also said he was “open” to the idea of a new relationship with the UK party, telling the Telegraph the “structure and direction” of the Scottish Tories should be examined.
But Mr Carlaw argued that the party already had its own “identity” and “distinct view” on issues north of the border, describing the idea of a split as “old”.
“I just don’t think an introspective debate of that character is going to impress people when there are huge issues facing the country at the present time,” he told the BBC.
“I do believe that we have that independence to develop our own political agenda on devolved issues – that was one of the prizes Ruth won.”
The interim leader, who may also enter the race to succeed Ms Davidson, said he accepted that a debate should take place eventually to discuss the future of the party in Scotland but not in the near future.
“Now is not a time for an internal discussion about the future of the Scottish Conservative party,” he added.
“We know we are at a critical moment in the country’s future, we have the Brexit debate and prospect, there could be a general election.
“At some point there will be a need for the party to have a proper examination of where it wants to go and how it wants to proceed.”
The SNP said the Tories were guilty of “enormous double standards” for considering a possible split from the UK party given their opposition to a second independence referendum.
“This is gross hypocrisy from the Tories, who are now demanding independence from their Westminster bosses at the same time as trying to block the people of Scotland being given a say on their future,” said the party’s depute leader Keith Brown.
“Boris Johnson’s government is too toxic for Ruth Davidson – and now it seems is too extreme for the Scottish Tories as a whole.”