Trump dismissed a suggestion by host George Stephanopoulos that by mentioning Carson's religious affiliation he was trying to "send a dog whistle" to "some conservatives [who] claim the Seventh-day Adventists are not Christian."
"No, not at all," Trump said.
Trump had brought up Carson's religion on Saturday, during a rally in Florida.
"I'm Presbyterian, he said. "Boy, that's down the middle of the road, folks, in all fairness. I mean, Seventh-day Adventist, I don't know about. I just don't know about."
Carson, in an interview on FOX News Sunday, declined to strike back but noted that Trump "went ballistic" several weeks ago when Carson questioned Trump's faith, saying the difference between him and Trump was that “I’ve realized where my success has come from, and I don’t in any way deny my faith in God.”
"So it seems a little interesting that he would now be doing that," Carson noted. Carson, who has also said he would not support a Muslim for president, later apologized for the comments about Trump's faith.
Carson also passed when Fox News anchor Chris Wallace reminded him of Trump's criticisms about his rival's energy level and his stand on immigration.
"I really refuse to really get into the mud pit," Carson said, adding that Trump "is who he is. I don't think that's going to change. And I am who I am. That's not going to change, either."
He continued: "And the way I kind of look at it, if people resonate what I'm talking about ... And if they like that, and it works with them, and they feel I'm the good representative for them, that's great. I would love to have their vote. And if they don't want me, that's fine, too. Because I would never lie just to get an office. I wouldn't be happy, and the people wouldn't be happy."
The Seventh-day Adventist Church has struggled historically to be accepted by mainstream Protestants. It traces its roots to lay theologian William Miller, who predicted the return of Jesus Christ in 1844, the movement was abandoned by some of its early followers and dismissed as a cult after Miller's prediction didn't pan out.
Over the decades, leaders of the church have worked hard to emphasize their belief in Christian doctrine and practices, though some Christians still look at them with skepticism, as they do with Mormons and Jehovah's Witnesses. Mitt Romney, the GOP's 2012 nominee, is a Mormon, and while his faith was mentioned, it was never a major issue in that race.
Carson, who has said that he was twice baptized as a Seventh-day Adventist, talks frequently about his belief in God and the need to defend religious freedom, but not as much about his specific faith. In a 1999 interview with the Religion News Service, he said: "I spend just as much time in non-Seventh-day Adventist churches because I’m not convinced that the denomination is the most important thing. I think it’s the relationship with God that’s most important.”
In their dueling Sunday show appearances, Trump kept up his aggressive rhetoric about his political prowess and Carson continued his confident but low-key approach to the campaign.
Trump suggested in interviews on Sunday that he was a bit taken aback by polls that show Carson, whom he has criticized as lacking energy, pulling ahead of him in Iowa. A Bloomberg Politics/Des Moines Register released Friday showed Carson with 29 percent support to Trump's 19 percent. The day before a Quinnipiac University poll had Carson ahead of Trump 28-20.
"I was really surprised to see it, because three nights ago, I was in Iowa. We had a packed house. We had 4,000 people, and it was a lovefest," Trump said in an interview on CNN's "State of the Union." "And I have done really well with the evangelicals and with the tea party and everything. And I just don't understand the number. But you know what? I accept the number. It means I have to work a little bit harder in Iowa."
Carson, in an interview on Fox News, attributed his surge in Iowa to "the power of social media and of word of mouth because as you know ... a lot of the media has it in for me. But, you know, if people listen to them, you know, I would be polling at less than zero."
Carson, who has gotten high marks for likability from grassroots Republican Party activists, continues to struggle to explain his policy positions. Wallace spent several minutes trying to get him to clearly articulate his plan for personal health-care accounts, which Carson has suggested could replace Medicare for some older Americans.
Wallace kept saying that Carson was not being clear about whether and how much the government would contribute to the accounts and whether such accounts are simply a new form of Medicare.
"If we take those same dollars and divert them into a system that gives you control over your home health care, you and your health-care provider cut out the middle man, the bureaucracy. Those dollars go much further. We won't have to use many of them. The dollars are already there, Chris," Carson said.
Wallace seemed unconvinced and ended by interview by saying: "Well, this is interesting, obviously, to be continued.... With more prominence in the polls, more discussion of your proposal."
Trump, in his "This Week" interview, said he agreed with Carson's proposal.
"I'm okay with the savings accounts. I think it's a good idea; it's a very down-the-middle idea. It works. It's something that's proven," he said.