"Clearly, any derogatory comments about anyone's faith -- those comments are troubling. The fact they keep on coming up is even more troubling" Romney said on Saturday.
The Mormon church is one of the fastest-growing religions and claims about 12.5 million members worldwide, but many evangelical Christians in key primary states such as Iowa and South Carolina consider the faith a cult.
In May, the Rev. Al Sharpton toured the headquarters of the Mormon church after comments he made in a debate were construed as a degrading to Romney's Mormon religion.
Romney's comments follow an apology from GOP rival John McCain's presidential campaign for comments about the Mormon church allegedly made by a volunteer earlier this year.
That apology came just days after Republican presidential hopeful Sam Brownback issued a similar one for a campaign staffer's e-mail to Iowa Republican leaders that was an apparent attempt to draw unfavorable scrutiny of Romney's religion.
Romney said in a large presidential campaign there will always be some volunteers or staff members who can't be controlled. But he said the difference between derogatory comments that originated from the McCain campaign and others is that the Arizona senator hasn't personally apologized to him.
"In the case of Senator Brownback and Mayor (Rudy) Giuliani ... they called immediately. They each spoke with me personally. I don't have any issue with that at all," Romney said. "(McCain) can do whatever he feels is the right thing. There's no need for me to suggest how people respond to things that go on in the campaign."
Romney, a former governor of Massachusetts, said he had not spoken with McCain since the last presidential debate on June 5.
Tucker Bounds, a McCain campaign spokesman, said the McCain campaign has already apologized.
"It's a very sincere apology. There is absolutely no place for those type of comments in our campaign," he said.
Romney used a fundraiser hosted by Utah Jazz owner Larry Miller to criticize the McCain-Feingold campaign finance bill. McCain-Feingold banned unregulated, unlimited "soft money" contributions from corporations, unions and wealthy individuals to national political parties and federal candidates.
"Governor Romney seems to forget that he supported campaign finance reform and even pushed for taxes to be paid on contributions. His unwillingness to address his past support campaign finance reform is part of a pattern that illustrates he will say and do anything in an attempt to win the nomination," Bounds said.
Romney said he favors unlimited donations as long as they are immediately disclosed on the Internet.
"The bill ought to be repealed," he said. "It's been the wrong course for American campaigns."
Romney was attending fundraisers in Salt Lake City and in Logan on Saturday.