Several Republican lawmakers, including Romney's presidential rival John McCain, have said the three-term senator should step down. The 62-year-old Craig pleaded guilty earlier this month to a charge stemming from an undercover police operation last June in a Minneapolis airport bathroom.
Craig said Tuesday he had committed no wrongdoing and shouldn't have pleaded guilty. He also said he was not gay.
Campaigning in South Carolina, Romney was asked by a voter why Republicans were distancing themselves from Craig.
"I think it's appropriate when there's been conduct that we think is disappointing and disgraceful to indicate that. I don't think there's a responsibility to try to gild the lily in a setting like this," said the former Massachusetts governor. "I think individuals across the country expect us to have the same expression that they feel, which is disappointment."
Later, Romney said he would not tell Craig what to do.
"I expressed my view that his conduct was disappointing, disgraceful, and at this stage, he has a decision of his own to make about his future. I'm not going to insert myself between him and his conscience," Romney said during a stop in North Charleston.
Craig was Senate liaison for Romney's campaign, a post he abandoned on Monday when the scandal came to light.
While in Myrtle Beach, Romney said Craig's problems won't become his own and that they were dealt with properly.
"People recognize that it's tough enough being a candidate and that one's supporters are going to sometimes make you look better than you deserve and at other times can be disappointing," Romney told a crowd of more than 80 at a Myrtle Beach hotel. "This is a setting where Senator Craig has been disappointing to his colleagues and to his citizenry."
Romney was touring the early voting state where Republicans plan to hold their primary on Jan. 19. He was using a bus dubbed the "Mitt Mobile."