Correction: A previous version incorrectly identified the country with which President Trump is re-negotiating a free trade agreement. It is South Korea.

SEOUL — President Trump said Tuesday he will continue to push South Korea and other nations across the globe to pressure North Korea into giving up its nuclear weapons, calling their existence unacceptable.

While suggesting he is willing to use military force if necessary, Trump also told reporters during a visit to Seoul that he sees "good progress" on hopes that North Korea will "make a deal" regarding their nukes; the president did not elaborate.

"North Korea is a worldwide threat that requires worldwide action," Trump said at a news conference following meetings with South Korea President Moon Jae In.

The president also said that he and Moon have agreed to re-negotiate the free trade agreement between the United States and South Korea, in order to make it "free, fair, and reciprocal."

While North Korea and trade have dominated his week-long trip to Asia, Trump also faced more gun control questions in the wake of Sunday's mass shooting a church in Texas.

Asked if prospective gun owners should undergo "extreme vetting," Trump said new gun control measures wouldn't help address the problem. The president noted that a legally armed man counter-attacked the church assailant, chasing him off and probably saving lives.

Trump also expressed annoyance that a gun control question would be asked in South Korea, where anxiety about North Korea nuclear tests — and about Trump's harsh comments toward North Korean leader Kim Jong Un — have triggered fears of war.

Trump said he would engage in military action "if need be," but also expressed confidence the impasse will be resolved as he engaged in meetings and ceremonies less than 50 miles from the North Korea border.

The president who has mocked Kim as "Rocket Man" and threatened to rain down "fire and fury" if he attacks the United States or its allies modulated his rhetoric during his visit to South Korea.

Ultimately, it will all work out; it always works out; it has to work out," Trump said during an earlier briefing at Camp Humphreys, a U.S.-South Korean military base near Seoul.

Moon and his aides, however, have expressed discomfort with Trump's aggressive rhetoric, though the South Korean president expressed general support for the American's efforts after their meetings on Tuesday.

As Trump headed toward Seoul, South Korea Foreign Minister Kang Kyung Wha said all officials need to be careful with any talk that smacks of war.

More:With North Korea threat looming, Trump seeks $5.9 billion for military

More:Trump visits Seoul, where 'fire and fury' comments on North Korea inflame protesters

"Another war on the Korean Peninsula must not happen," Kang told NBC News. "A resolution to the North Korean nuclear issue must be pursued in a peaceful, diplomatic manner."

Moon and his government have advocated a more diplomatic approach based on multi-lateral talks with North Korea.

Trump, meanwhile, has described Moon as an "appeaser," though the two leaders made a show of getting along Tuesday.

During a bilateral meeting at the "Blue House," South Korea's presidential mansion, Moon played up to Trump by noting that Wednesday is the one-year anniversary of his election, and "you are already making great progress on making America great again." (Trump replied: "It was a great victory, and a victory that made a lot of people very happy.")

As for North Korea's threats, Moon said he hopes Trump's visit could be a "turning point" in the impasse, and "relieve some of the anxiety that the Korean people have due to North Korea's provocations."

In responding comments, Trump did not mention North Korea, but praised the people of Asia in general. “There is a great spirit, it’s a great part of the world," Trump said.

Trump picked Seoul as the site for a major part of his Asia trip, a Wednesday speech to the South Korea National Assembly on his efforts to forge a regional coalition to oppose North Korea.

The American president arrived in South Korea bearing a gift: A new request to Congress for military spending that includes $4 billion for a missile defense and detection system on the Korean peninsula.

Moon and other South Korean officials have been leery of an American-backed missile defense system, in part because of objections from China. The Chinese say a missile defense system could be used as an offensive system against them.

Trump wants countries in the region to pressure Kim over his weapons, especially China, North Korea's neighbor and largest economic patron. He also wants help from Russia.

As part of his Asia tour, the president will visit Beijing later this week to speak with President Xi Jinping about North Korea and other issues.

The trip also will take Trump to economic conferences in Vietnam and the Philippines. He will speak with Asian leaders about possible plans to change trade laws with the United States.

After landing in South Korea, Trump headed to Camp Humphreys, the American-South Korean military where he had lunch with American and Korean troops and received his military briefing. The president spotlighted the camp as an example of cooperation and burden-sharing between the U.S. and its allies.

The president and first lady Melania Trump then headed to Seoul for a welcoming ceremony at the Blue House.

Trade is also high on Trump's agenda. The president has criticized aspects of the current U.S. free trade agreement with South Korea and said he would discuss his concerns with Moon. "It will start working out and working out so we create lots of jobs in the United States, which is one of the many important reasons I am here," Trump told reporters.

Meanwhile, hundreds of demonstrators gathered in downtown Seoul to protest Trump's visit, saying his bellicose rhetoric makes war with North Korea more likely. Some waved red signs that said, "No Trump, No War."

North Korea's state-run media denounced Trump as he made his way through Asia, and warned other countries — read: South Korea — not to cooperate with him.

“Nobody can predict when Trump does a reckless act," one state-run newspaper said. "The only and one way for checking his rash act is to tame him with absolute physical power. We warn Trump's coteries once again. If they want to get rid of ruin, do not make reckless remarks.”

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