In an interview with Beliefnet, a multi-denominational Web site that covers religion and spirituality, the White House hopeful was asked if a Muslim candidate could be a good president.
"I just have to say in all candor that since this nation was founded primarily on Christian principles ... personally, I prefer someone who I know who has a solid grounding in my faith," McCain said. "But that doesn't mean that I'm sure that someone who is Muslim would not make a good president."
Later, McCain said, "I would vote for a Muslim if he or she was the candidate best able to lead the country and defend our political values." He added that "the Constitution established the United States of America as a Christian nation."
The interview was published Saturday.
The American Jewish Committee, an international think tank and advocacy organization based in New York, issued a statement criticizing the Arizona senator, arguing that McCain should know that the United States is a democratic society without a religious test for public office.
"To argue that America is a Christian nation, or that persons of a particular faith should by reason of their faith not seek high office, puts the very character of our country at stake," Jeffrey Sinensky, the group's general counsel, said Monday in a statement.
A partisan organization, the National Jewish Democratic Council, also called McCain's comments repugnant.
Amid the criticism, Democrat Joe Lieberman of Connecticut, an Orthodox Jew, came to the defense of his Senate colleague.
"I have known John McCain very well for many years and I know that he does not have a bigoted bone in his body. I know that he is fair and just to all Americans regardless of their faith," Lieberman said.
Over the past few days, McCain has sought to clarify his remarks.
While campaigning in New Hampshire on Sunday, he said that the most qualified person could be president, no matter his or her religion.
"It's almost Talmudic. We are a nation that was based on Judeo-Christian values. That means respect for all of human rights and dignity. That's my principle values and ideas, and that's what I think motivated our founding fathers," McCain said.
Also Sunday, in a statement, his spokeswoman Jill Hazelbaker said: "The senator did not intend to assert that members of one religious faith or another have a greater claim to American citizenship over another."