Diplomats fretted that Trump was making a bad situation worse by threatening military action. North Korea said it is considering how to respond and suggested Friday that it may soon test a hydrogen bomb over the Pacific.
The latest exchange began late Thursday, when Kim called Trump a "mentally deranged U.S. dotard" and referred to Trump's speech at the United Nations on Tuesday as "unprecedented rude nonsense."
Name-calling is standard procedure for North Korea, but the kind of direct statement issued in Kim's name Thursday night is unusual.
Trump responded early Friday via Twitter.
"Kim Jong Un of North Korea, who is obviously a madman who doesn't mind starving or killing his people, will be tested like never before!" the president tweeted.
Trump returned to the theme later Friday, saying at a political rally in Alabama that "Rocket Man should have been handled a long time ago."
"I want to tell you something, and I'm sure he's listening because he watches every word. He's watching us like he never watched anybody before. And maybe something gets worked out and maybe it doesn't. Personally, I'm not sure it will," Trump said. "But I'll tell you one thing: You are protected."
The Trump administration said last week that Trump is merely using the kind of blunt language that the isolated North Korean leader understands. But officials acknowledge that they know little about how Kim perceives the world.
Asian diplomats have warned the United States that the risk is particularly acute if Kim feels he is losing respect with the North Korean military and government elite.
Pentagon leaders also worry that in the current charged environment, either leader or his military might misinterpret the other's intentions. So far, the White House has pursued a sometimes confusing three-part strategy of tougher economic sanctions, military threats and an offer of negotiations.
"We don't know how their means of communication and behavior will be," Secretary of State Rex Tillerson said Wednesday. "How the decision-maker or the people that are closer to the decision-maker are going to behave is something we will have to understand and learn."
Trump declared Tuesday while addressing the U.N. General Assembly that "the United States has great strength and patience, but if it is forced to defend itself or its allies, we will have no choice but to totally destroy North Korea."
North Korea's official Korean Central News Agency on Friday referred to Trump's speech as "rubbish." "This thrice-cursed sophism made by the mentally deranged hooligan has shocked the whole world," the news agency said.
The exchange of insults came as China on Friday disputed Trump's claim that it has joined a new round of punishing economic sanctions against its communist ally.
Unveiling the penalties Thursday, Trump said that Chinese President Xi Jinping had ordered Chinese banks to cease conducting business with North Korean entities. He welcomed the move as "very bold" and "somewhat unexpected."
But on Friday, China's Foreign Ministry spokesman denied that Beijing had agreed to go that far.
"As far as I know, what you have mentioned just now is not consistent with the facts," spokesman Lu Kang said at a regular news conference in response to a question about Trump's comments.
Rising tension and the threat of a new war on the Korean Peninsula shaped Trump's four days of diplomatic discussions at the United Nations. He met with the leaders of Japan and South Korea — U.S. allies directly threatened by North Korea — and mocked Kim as "Rocket Man" on a "suicide mission" before announcing sweeping new U.S. financial sanctions.
Kim did not attend the session, and neither did Xi or Russian President Vladimir Putin, whose nation also does business with North Korea.
Russian Foreign Minister Sergei Lavrov warned the gathering Thursday of "a very dangerous confrontation spiral."
"We resolutely condemn the nuclear missile adventures of Pyongyang in violation of Security Council resolutions. But military hysteria is not just an impasse, it's disaster."
Trump also said diplomacy can still work to avert a crisis, and U.S. officials scurried to tell allies that Trump is not itching for war.
When asked Friday whether the president's tweets were working to avert war, Tillerson highlighted the U.S. diplomatic efforts to address the North Korea situation.
"The president obviously takes the responsibility of the security of the American people very seriously, and it is his first and foremost responsibility," Tillerson said on ABC's "Good Morning America."
"We will continue our efforts in the diplomatic arena, but all of our military options — as the president has said — are on the table," he added.
The White House said Friday that Trump and South Korean President Moon Jae-in, who met Thursday, "shared the view that maintaining overwhelming military superiority over North Korea is essential."
China has tightened the financial screws on North Korea under U.N. sanctions approved in recent weeks. But Beijing remains unwilling to completely isolate the regime and has not cut all financial ties.
Given that China accounts for nearly 90 percent of North Korea's trade, Beijing's cooperation is vital to Trump's efforts to isolate and pressure Pyongyang.
The executive order Trump issued Thursday imposes sanctions on any foreign financial institution that knowingly conducts or facilitates "any significant transaction in connection with trade with North Korea." It also bans aircraft and ships from entering the United States if they have traveled to North Korea in the preceding 180 days.
At Friday's Chinese Foreign Ministry briefing, Lu repeated Beijing's familiar talking points: that China "comprehensively and strictly" implements U.N. resolutions but opposes unilateral sanctions imposed outside the U.N. framework.
"China's stance on this is clear and consistent," he said.
Lu declined to comment on the "personalized" exchange between Trump and Kim.
He also, again, appealed for calm.
"What's needed now is to implement the U.N. resolutions strictly and positively explore channels to solve problems via talks, rather than provoking each other and adding oil to the fire," Lu said.
Denyer reported from Beijing. Anna Fifield in Tokyo; Shirley Feng, Luna Lin and Liu Yang in Beijing, and Abby Phillip in Huntsville, Ala., contributed to this report.