James Heappey is seeking re-election in Wells, Somerset, and made the remark as a “joke” on a visit to the county’s Millfield School.
He asked sixth-form pupils at the £12,000-a-year private school their opinion on independence.
When one Scottish student told him she supports independence, he reportedly told her to “f*** off back to Scotland”.
She complained to her father and the school sent a letter of apology to the family.
Mr Heappey said the comment was a “joke” and he has written to the pupil to apologise for causing any offence, but the SNP wants him to be suspended from standing in the General Election.
First Minister Nicola Sturgeon told Sky News: “I think it’s appalling behaviour for any adult to show to any young person, but particularly somebody who has been an MP and is standing for election again.
“I think the Tories really have some big questions to answer about the conduct of the some of their candidates and elected representatives.”
SNP candidate for Glasgow North West Carol Monaghan, said: “For a candidate to make this kind of remark, especially to a schoolgirl, is absolutely disgraceful and the Tories should suspend Mr Heappey as a candidate.
“Mr Heappey’s protests about it being a joke just don’t cut it. This sort of offensive and abusive language has no place in our politics.”
Alex Cole-Hamilton, chairman of the Scottish Liberal Democrat election campaign, called on Scottish Conservative leader Ruth Davidson to “immediately disown the actions” of Mr Heappey.
He said: “Once again a senior Conservative manages to help the SNP by being obnoxious.
“Scots who oppose a divisive independence referendum should vote for the Liberal Democrats because we can manage to oppose independence without swearing at children.”
Mr Heappey said in a statement: “I made a comment - intended only as a joke - but it was inappropriate and I am deeply sorry for any offence caused.
“I wrote to [the pupil] soon after the school brought her concerns to my attention and apologised unreservedly.”