Updated at 12:03 p.m.
The South Carolina senator was asked whether the official GOP platform should include an endorsement of such an amendment, as it did in 2012. His response: It's not a good idea.
"You can put it in the platform, but it will, in my view, hurt us in 2016, because it's a process that's not going to bear fruit," said Graham.
Graham said that in the wake of Friday's landmark Supreme Court ruling legalizing gay marriage nationwide, his goal is to "protect the religious liberties" of those who oppose gay marriage as part of their faith.
Louisiana Gov. Bobby Jindal, another Republican candidate for president, reiterated his strong disagreement with the court's ruling, but said his state, which has not implemented the change in policy yet, will abide by it.
"Our agencies will comply with the court order," he said on "Meet The Press."
Human Rights Campaign president Chad Griffin, looking ahead, said that while Friday's ruling was a "monumental leap forward," work remains to be done.
"We still have a long ways to go," he said on CBS's "Face The Nation." "You know, in a majority of states in the country, still, today, after this ruling, you can be married at 10 a.m., fired from your job by noon and evicted from your home by 2, simply for posting that wedding photo on Facebook. And so, as you look at the battles ahead, we've got to bring full and comprehensive nondiscrimination protections to everyone living in every state in this country, and that's the next battle in Congress."
Russell Moore of the Southern Baptist Convention said on the same program that demands for specific kinds of religious freedom protections will depend on what gay rights advocates do next.
"Are they going to take a French Revolutionary approach to the sexual revolution or not?" he said. "And I think we have to hold by our First Amendment protections for people of faith and for religious institutions because we're not going to be able to simply put our convictions in a bind trust."