Mike Russell, the Scottish Constitutional Relations Secretary, added that, while “everybody will have an opinion” on the timing of another Scottish referendum, he trusts the First Minister on the issue.
He spoke out after the SNP leader accused Mrs May of “running scared” of a second vote on independence.
Speaking after talks with the PM in Downing Street on Wednesday, Ms Sturgeon said: “I think the Prime Minister fears that she would lose another Scottish independence referendum so she’s running scared of the verdict of the people.
“It’s probably the reason she doesn’t want another EU referendum, because she thinks she might lose that. People that are confident in their arguments don’t run away from the verdict of the people.”
But Mrs May insisted it is the SNP that is “out of touch” with the people of Scotland, who she said do not want “another divisive Scottish independence referendum”.
The two leaders clashed at the same time as Ms Sturgeon faced growing calls within her own party to call for a fresh vote on Scotland’s future place within the UK.
Former first minister Alex Salmond has argued that there is “not likely to be a better time to force the issue” following the PM’s Brexit deal defeat.
Similarly, SNP Western Isles MP Angus MacNeil has argued that it is time for calls for a fresh independence vote “to come to fore”, urging Ms Sturgeon to prioritise this over demands for a second European referendum.
However, SNP MEP Alyn Smith said supporters of holding another independence vote in the near future should “ca’canny” as uncertainty over what will happen with Brexit means practical questions over independence remain unanswered.
Mr Russell told BBC Radio Scotland’s Good Morning Scotland programme that “everybody will have an opinion” about the timing of a possible second referendum.
He added: “There are clearly many interests to be taken into account and I absolutely trust the First Minister’s judgment on these things.”
Ms Sturgeon has already pledged to set out her views on this in a “matter of weeks”.
But with the UK facing a Brexit “crisis” which he said was “getting considerably worse”, Mr Russell said Scots should “look to ourselves and our own future”.
He stated: “It is absolutely the right moment to say that these are very, very serious concerns we are facing and the people of Scotland have the right to say ‘Do we have to go through this? It is something that we did not vote for’.
“That is the truth at the heart of this. Scotland did not vote for Brexit, we’ve spent two-and-a-half years trying to negotiate the best possible outcome of this, compromising all the way, and we are incredibly within weeks of the supposed end date without anything firm in place.
“Now the people of Scotland have the right to say ‘That is unacceptable, that is damaging and we must look to ourselves and our own future’.”
Following Wednesday’s meeting between the First Minister and Mrs May, Downing Street revealed that both Ms Sturgeon and her Welsh counterpart, Carwyn Jones, had been invited to sit on a new Cabinet sub-committee which will “work on preparations for the UK’s exit from the EU, spanning both deal and no-deal outcomes”
But Mr Russell said he had previously found promises of more involvement and consultation from the UK Government “to be hollow”.