Ms Sturgeon and her fellow nationalists sent a warning to No 10 after the defence secretary said the Government would not give Holyrood the power to stage another referendum.
Michael Fallon said he “didn’t see the need for a second referendum” and called for Ms Sturgeon and the SNP to “get on with the day job”.
Speaking during First Minister’s Questions on Thursday Ms Sturgeon was asked by MSP Tom Arthur: “Would the First Minister agree that if this parliament chooses to have a referendum on Scotland’s future, then no Westminster Tory should try and stand in the way.”
In reply, Ms Sturgeon, who has continually threatened a second independence vote if her wishes over Brexit are not answered, said the Conservatives were frightened of another referendum on Scotland’s future.
She said: “If this parliament voted to have a referendum on independence then no Westminster Tory should stand in the way of the voice of this parliament.
Nicola Sturgeon hit back after Michael Fallon moved to shut down a second independence referendum
“The mandate of this government in relation to this is unequivocal it was the Tories after all who put us in the position of being taken out of the European Union against our will and with the support of just one of the 59 MPs in this country.
“Strange is it not that a Tory Party which proclaims that it would be so confident of winning a referendum on independence now talks about trying to block it.”
When asked if the UK Government would facilitate another referendum in this Parliament, Sir Michael told the Herald newspaper: "No, forget it. The respect agenda is two-way. She (the First Minister) is constantly asking us to respect the SNP Government but she has to respect the decision of Scotland to stay inside the UK in 2014 and the decision of the UK to leave the EU.
"Respect works two ways."
Ms Sturgeon accused him of "backpedalling" on the issue when he later refused to say if Westminster would block a second independence referendum.
On a visit to naval engineering firm MacTaggart Scott in Loanhead, Midlothian, he said talk of a referendum is "irrelevant" and accused the First Minister of trying to "divert us from her failure".
Michael Fallon said the Tories would look to block a second independence vote
If this parliament voted to have a referendum on independence then no Westminster Tory should stand in the way of the voice of this parliament
Pressed if the UK Government would block a second independence referendum, he said: "I think the Scottish Government should focus on what it was elected to do and that's jobs, school standards, problems in the health service, that's what people should be focusing on rather than theoretical talk about a referendum.
"I've made it very clear that there's no need for another referendum at the moment. There's plenty for the Scottish Government to do, to be backing successful companies like this, to be sorting out problems in the Scottish health service, to be dealing with falling school standards.
"Nicola Sturgeon is trying to distract us all from the problems she's having in getting to grips with what she was elected to do, which is to deal with the Scottish economy. Why is unemployment still growing in Scotland and still falling in England?
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Nicola Sturgeon visits Glaxo Smith Kline.
"All this talk of a referendum, she is trying to divert us from her failure to grip the issues she should be dealing with, she needs to get back to the day job."
Ms Sturgeon tweeted "serious backpeddling", adding "must have realised what a disastrous move it would be for Tories".
A spokesman for the First Minister added: "Any Tory bid to block a referendum would be a democratic outrage, but would only succeed in boosting support for both a referendum and for independence itself – something which the Prime Minister has previously indicated she understands all too well.