Insurgent Democrat presidential candidate Pete Buttigieg acknowledged at the Fox News town hall in Claremont, New Hampshire Sunday evening that he needs to do more to appeal to "black and brown" voters, even as he confidently parried a series of policy questions -- and, on several occasions, went directly after President Trump.

Buttigieg argued that minority voters are "skeptical of people who seem to come out of nowhere," after moderator Chris Wallace noted that he was polling at one-percent support among nonwhite primary voters according to a recent Fox News poll.

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On fiscal policy, Buttigieg pushed for four distinct tax hikes when asked about the deficit, saying he favored a "fairer, which means higher" marginal income tax, a "reasonable" wealth tax "or something like that," a financial transactions tax, and closing "corporate tax loopholes."

"You don't blow a hole in the budget with an unnecessary and unaffordable tax cut for the very wealthiest," Buttigieg told Wallace, referring to President Trump's tax legislation.

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With little equivocating, Buttigieg largely stuck to reminding voters of his core campaign pledges, and the lessons he learned from his six-month deployment to Afghanistan in 2014: "We do not send young men and women to war when there's an alternative," he emphasized.

Throughout, Buttigieg repeatedly drew applause -- even when he defended his calls to abolish the Electoral College, a move that would gut New Hampshire's influence in selecting the next president.

"States don't vote, people vote," Buttigieg said. He added that "if we're going to call ourselves a democracy," the U.S. should move to a popular vote system.

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Responding to the newly passed pro-life legislation passed in Alabama, as well as similar bills making their way through other state legislatures, the 37-year-old mayor of South Bend, Indiana told Wallace that "abortion is a national right."

Asked about third-trimester abortions, Buttigieg asserted that such cases were rare and typically involved an unforeseen circumstance, before refusing to endorse any restrictions whatsoever on late-term abortions. Government statistics indicate that in 2015, approximately 1.3 percent of abortions were performed after 20 weeks.

"I trust women to draw the line," Buttigieg said. "That decision [to have an abortion] is not going to be made any better, medically or morally, because the government is dictating how that decision should be made."

Vice President Joe Biden is now the clear front-runner in the crowded Democrat Party primary field, but Buttigieg indicated that the long odds didn't faze him.

"There's a lot of us running for president on the Democratic side, but I think it's safe to say I'm not like the others," Buttigieg told Wallace, noting that seeking the presidency is inherently "audacious" -- especially given that he would be the youngest person to ever become president.

"I would say being a mayor in a city of any size in America right now is about as relevant as it gets," Buttigieg added.

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Buttigieg suggested he isn't too worried about his likely Republican opponent after the primary, either.

"The tweets are --- I don't care," Buttigieg to raucous applause at one point, referring to Trump's Twitter posts, including some that mocked Buttigieg.

"It's a great way to command attention," Wallace said.

"It's a great way to command attention from the media," Buttigieg countered.

In a closing lightning-round of questions, Buttigieg downplayed his comments during a radio interview on Friday, in which he called for removing Thomas Jefferson's name from buildings and events like the Democratic Party’s annual Jefferson-Jackson dinner.

Jefferson, the primary author of the Declaration of Independence who also served as both president and vice president, has become the latest historical figure that progressives have pushed to "disappear" because of his slave ownership.

"You would have thought I would proposed blowing up the Jefferson Memorial in D.C.," Buttigieg joked, noting that his campaign headquarters is on a street named for Jefferson. Buttigieg reiterated that he did not want Jefferson's name stripped entirely from the public square, but instead from honorary titles.

Buttigieg also cited the HBO show "Game of Thrones" as one of his "guilty pleasures," noting that the finale would be televised shortly after the conclusion of the town hall.

Ultimately, Buttigieg received a standing ovation after delivering his closing remarks.

"Look, what we’re trying to do here is different," Buttigieg began, as the the town hall wrapped up. "Because the moment that we’re in is different. I get that a millennial, midwestern mayor is not what leaps to mind when you think about a prototypical candidate for president. But I also think we’re living — if it’s hard to figure out what’s going on right now, it’s because we are living on one of those blank pages in between chapters of American history. And what comes next could be ugly or it could be amazing."

Buttigieg continued: "And I believe running for office is an act of hope, and so is voting for somebody, and supporting somebody and volunteering for somebody. I hope you’ll join me in making sure that that next era is better than any we’ve had so far."

Before the town hall, Buttigieg took stock of his campaign's unexpectedly strong position early in the race.

“We were expecting at this stage of the game to still be introducing ourselves and even defending the idea that something this audacious was appropriate,” Buttigieg told Fox News. “Instead we find that we’ve bolted into the top tier.”

In an email to supporters Saturday night, Buttigieg defended his decision to appear at the Fox town hall, as progressives have increasingly pushed for Democrats to appear only on left-leaning networks. On Sunday, Trump himself took aim at Buttigieg's appearance at the town hall, saying it was "hard to believe" Fox had hosted the candidate.

“If we ignore the viewers of Fox News and every news platform that doesn’t share our worldview, we will surrender our ability to speak directly to millions of American voters,” Buttigieg asserted. “If we don’t show up, the conservative media will tell our side of the story for us.”

Separately, Trump told Fox News' "The Next Revolution with Steve Hilton" in an interview to air Sunday that "I think it's absolutely fine" that Buttigieg and his husband, Chasten, are open about their relationship on the campaign trail.

Trump agreed with Hilton that Buttigieg's candidacy is a sign of progress for the country.

"I think that's something that perhaps some people will have a problem with," Trump said. "I have no problem with it whatsoever. I think it's good."

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During his 2016 campaign, Trump suggested he'd appoint conservative justices to overturn a Supreme Court decision legalizing same-sex marriage. But he called the issue "settled" shortly after his election.

Fox News' Paul Steinhauser and The Associated Press contributed to this report.