While some saw the President's phrasing as a clear message of deterrence, others said it may deepen confusion about what exactly will prompt the US to execute a preemptive military strike.
The spiraling tension between the North Korea and the US moved at least two world leaders at the UN to urge caution and greater care with language, suggesting that some see the US as part of the problem.
"When tensions rise, so does the chance of miscalculation," said UN Secretary General Antonio Guterres, who urged the DPRK to comply with UN resolutions and condemned its nuclear tests, the sixth of which took place just weeks ago.
"Fiery talk can lead to fatal misunderstandings, we must not sleepwalk our way into war," Guterres warned as he opened the assembly. He was echoed by French President Emmanuel Macron, who spoke after Trump.
"Our responsibility to all of our partners, including China and Russia, is through resolve to bring North Korea to the negotiation table for a political settlement to this conflict," Macron said. "France rejects escalation." Later, at a press conference, he underscored the point, saying: "Any military intervention must be avoided."
Macron was speaking after Trump had taken the podium and warned that if the US "is forced to defend itself or its allies, we will have no choice but to totally destroy North Korea." He added, "the United States is ready, willing and able. But hopefully, this will not be necessary."
Referring to leader Kim Jong Un with an apparent reference to a 1972 Elton John song, Trump said that "Rocket Man is on a suicide mission for himself and for his regime."
Reaction in UN corridors after the President's remarks was "mixed," "at best," according to one UN official, while another senior UN diplomat said there was shock in the room when Trump threatened destruction.
"You could feel a wind had gone into the room when he said that," the diplomat said. "People were taken aback. There were rumblings." This official says it was the first time in his memory that a world leader has called for the obliteration of another state at the UNGA, noting even Iran's most fiery leaders didn't similarly threaten Israel.
The threat is likely to ratchet up tensions with North Korea while doing little to reassure US allies in Asia, said analysts who added that the President now also runs the risk of appearing weak if he doesn't follow through.
Trump was speaking as the US and its allies have sharply picked up the pace of operations in and around the Korean peninsula in answer to Pyongyang's ongoing ballistic missile and nuclear tests.
Japan has deployed a missile interceptor near the flight path of North Korean missiles, two of which have overflown the country. Troops from the US and South Korea, meanwhile, staged live fire drills Tuesday, just 20 miles from the border with North Korea. Their drill, dubbed "Warrior Strike 8," was a mock battle to retake a village from enemy hands.
US rhetoric has intensified too, with Defense Secretary James Mattis telling reporters -- without offering specifics -- that the US has military options to deal with North Korea that wouldn't put Seoul at risk.
And other Pentagon officials have told CNN's Barbara Starr that a debate is now underway about whether North Korea's missile program has reached the level of threat that the Pentagon would recommend targeting a missile, even if its trajectory did not indicate it would hit the US or its allies. The official declined to speak on the record because of the sensitivity of the issue.
Speaking to reporters Monday, Mattis said North Korea is "intentionally doing provocations that seem to press against the envelope for just how far can they push without going over some kind of a line in their minds that would make them vulnerable."
But he's also said diplomacy is the administration's leading option. "We are dealing with the North Korea situation through the international process and we will continue to do so," Mattis told reporters Tuesday. Secretary of State Rex Tillerson "is leading the effort and we will hopefully get this resolved through diplomatic means," he said.
Some analysts said Trump's tough talk about "total destruction" sends an important message.
"Clear deterrence signals like this are important for allied confidence and for avoiding any North Korean miscalculation stemming from the belief we are cowed into submission," said Victor Cha, a senior adviser at the Center for Strategic and International Studies.
But others, including Bruce Klingner, a senior fellow at the Heritage Foundation, said the Trump administration "has issued conflicting signals as to whether it would consider a preventative military strike on North Korea even without indications of imminent regime attack on the US and its allies."
Klingner said that previous comments by Trump, Tillerson, and National Security Advisor H.R. McMaster "all suggested the potential for such an attack," while Mattis' comments, "though strong and resolute, were in the context of a US response to a North Korean attack."
He notes that Trump's UN comments about North Korea's total destruction "appeared to be in response to a North Korean attack."
On top of confusion about what might prompt the US to strike, Carla Anne Robbins, an adjunct senior fellow at the Council on Foreign Relations, said "given how volatile things are on the Korean Peninsula right now," Trump's comment about destruction is "more likely to wind up Pyongyang, than it is to calm down Japan or South Korea."
Senator Diane Feinstein, a Democratic critic of Trump's from California, said the comment will also likely serve to isolate the US.
"Trump's bombastic threat to destroy North Korea and his refusal to present any positive pathways forward on the many global challenges we face are severe disappointments," she said in a statement. "He aims to unify the world through tactics of intimidation, but in reality he only further isolates the United States."
Robbins said she "cringed" when she heard the term "Rocket Man."
"Tweet language is beneath the dignity of the President of the United States," she said, adding that it will do little to encourage US allies in Asia will be taking diplomatic "risks and making sacrifices, which is inevitably going to have to happen ... as well as taking military risks" in order to follow US policy.
Dan Drezner, a professor of International politics at Tufts University's Fletcher School of Law and Diplomacy, said there was little benefit to "Trump's rhetorical flourishes, by which I mean, putting his foot in his mouth."
Referring to the Rocket Man reference as well as the threat to total destroy North Korea, Drezner said that "all its going to do is freak the media out; it's not going to have any effect on North Korea, except to make us look impotent when we don't follow through."
He points to other occasions when Trump has issued blistering threats -- for instance, in August when he promised "fire and fury" if North Korea threatened the US or its allies, and then failed to deliver a few days later when Pyongyang actually threatened Guam.
"He's bombastic and then the actual follow-through is weak," Drezner said.