Gingrich, who was on Fox's payroll as an analyst before running for president, said recently that if he was the GOP nominee, he wouldn't agree to a debate with President Barack Obama if a journalist was the moderator.
"I don't think that would work," said Baier, who has moderated five GOP debates this election cycle. "I don't think it would be too enjoyable to watch."
Baier, 42, has increased his profile with the debate work and status as co-anchor with Megyn Kelly on Fox's political night coverage. The nightly newscast he anchors, "Special Report," is the third most-watched news show on cable television, behind Fox teammates Bill O'Reilly and Sean Hannity, and it beats the combined audience of CNN and MSNBC at 6 p.m. Eastern. A former White House and Pentagon reporter for Fox, Baier took over "Special Report" from Brit Hume after the 2008 election.
Without the journalists, the debates would likely amount to little more than stump speeches, Baier said. Getting politicians off their programmed responses is the biggest challenge for debate moderators and usually produces the best moments.
But the Fox host understands where Gingrich is coming from.
"It's just politics," he said. "A lot of politicians have complained about media coverage and media questions. He just does it more frequently than others and perhaps more effectively."
Baier's not-so-tender moment with Romney came during a Nov. 30 interview. In a style he admired in the late Tim Russert, Baier confronted Romney with some quotes from the past that appeared to contradict what the candidate had been saying during the campaign. He asked: "How can voters trust that what they hear from you today is what you will believe when you're in the White House?"
Off-air later, Romney told Baier that he thought the interview was overly aggressive and that he didn't like it.
Romney's unhappiness was evident on the air, too. The unspoken subtext seemed to be: I thought Fox would be a friendlier venue than this.
"The news operation at Fox has established itself with a lot of folks, including people inside the GOP, as being fair and tough," Baier said. "There shouldn't be any surprise that we were asking questions that probably any other news outlet would want to ask."
Baier's audience skews right, as it does for most Fox shows. Forty-one percent of his audience identifies itself as Republican, 44 percent as independent and 15 percent as Democratic, according to a 2011 study by GfK MRI, a consumer research company.
The Washington-area think tank Center for Media and Public Affairs studied evening news coverage of the GOP campaign, including "Special Report," and concluded the Fox show's coverage was most balanced between positive and negative evaluations of the candidates. ABC and NBC were more negative while CBS, mostly due to a lengthy story examining Ron Paul's appeal, was more positive, the center said.
The news portion of Baier's show more resembles a meaty, old-style broadcast evening newscast than some of the broadcasters do today, said Robert Lichter, director of the center, based at George Mason University, and a communications professor. (Fox dominates its cable competition in the ratings although its audience is typically about a quarter that of NBC's "Nightly News.")
"He's a serious journalist," said Lichter, who was a part-time paid analyst at Fox until 2008. "He is in the tradition of Brit Hume as someone who tries to be fair and is someone who is worlds away from the talk-show hosts that really give Fox its brand."
His program's content does lean conservative in its second half-hour, Lichter said. That part of the show is dominated by a discussion Baier leads with a panel of three analysts. Generally, two of the three are conservative: Frequent contributors include syndicated columnist Charles Krauthammer and Stephen Hayes of The Weekly Standard.
The attention generated by something like the Romney dust-up can be positive because it can encourage people to sample his show, Baier said.
"The most vocal critics of Fox, most of them have not watched our news shows," he said. "I say to them, give me three days. Watch my show. Watch Shepard Smith's show."
Moderating the GOP presidential debates has been an education for Baier, who had never done that duty before on a national-level campaign. Mostly, it involved learning to avoid open-ended questions that candidates could slip away from and use as a launching pad for speeches.
"I do think that people get fed up with the back and forth, the day-to-day attacks," he said. "But that's part of politics as we know it now. I do think there's a hunger for substance. I actually think that's one of the reasons our show succeeds."
He backs up CNN's John King, who was angrily denounced by Gingrich for opening a CNN debate with a question about the former speaker's ex-wife's claim that he had asked her for an open marriage. Baier said he probably would have framed the question differently, putting it in the context of presidential character.
Given that the issue was dominating coverage on that day before the South Carolina primary, the topic was unavoidable, he said.
"The people who say, 'You just don't ask that question at all,' I don't think that's reality," Baier said.
Baier doesn't expect an early resolution to the GOP nomination fight. The debates have been good business for the cable news networks, less so on the nights turned over to coverage of primary or caucus coverage. It's all good for a political show like "Special Report," however.
"Brit built this foundation," Baier said. "We're just trying to complete the house."
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