But Alister Jack warned that a majority in alliance other pro-independence parties like the Greens - which currently exists at Holyrood - would not be enough to legitimise another vote on leaving the UK.
The claim appears to contradict Prime Minister Boris Johnson who insisted a fortnight ago on a trip to Scotland that he would not grant a section 30 order - allowing a second vote on independence - even if the SNP wins a majority in the the 2021 Scottish elections.
Ms Sturgeon has claimed that if the SNP wins the majority of Scotland's 59 seats on December 12 in the general election, it would give the party a mandate for a second referendum on independence.
But Mr Jack said this would not be the case. "The democratic mandate for a section 30 order is a matter for 2021," he told BBC Radio Scotland's Good Morning Scotland today.
"We'll see whether the Scottish National Party get a majority then - and I mean the Scottish national Party. Not in collaboration with other parties, not in any alliances, but a Scottish National Party majority which is what Ruth Davidson pointed out some months ago."
Mr Jack, who replaced David Mundell as Scottish Secretary when Mr Johnson became Prime Minister, made it clear that the Tories are opposed to another vote on Scotland leaving the UK.
"He's (Mr Johnson) made it very clear that the decision - the once in a generation decision - was taken in 2014. He has made it very clear that we do not support a second independence referendum.
"We believe that decision was taken. We should take away the uncertainty as we believe we should with Brexit, remove the uncertainty of independence and start growing the Scottish economy."
Power over the constitution in the UK lies with the UK Government. A second referendum would require a transfer of power, through a section 30 order, for it to take place.
The current pro-independence majority at Holyrood - combining the SNP and Greens - has already passed a motion demanding a second referendum.
A spokesman for Ms Sturgeon said today: "There is of course already a mandate, flowing from the 2016 election and the crystal clear manifesto commitment.
"More generally, I would just say that it shows Boris Johnson's position has been seriously undermined by Alister Jack in a way that shows their position is as flakey on the issue as Labour's.
"Nobody seriously believes that it is possible to stand in the way of a democratic mandate indefinitely. The position isn't sustainable and that will prove to be the case.
"So Alister Jack on one level is just stating what everybody knows - there is going to be another independence referendum."
Labour leader Jeremy Corbyn has indicated he could allow a second referendum in the later years of a Labour Government, possibly in deal with the SNP which would see nationalist MPs pave the way for him to become Prime Minister in a hung Parliament.