Trump responded by tweeting that Kim is "obviously a madman who doesn't mind starving or killing his people".
But North Korea's series of missile tests and apparent testing of a hydrogen bomb provoked Trump to increase his own insults and threats.
He offered more vitriol for Trump, saying he was "unfit to hold the prerogative of supreme command of a country, and he is surely a rogue and a gangster fond of playing with fire, rather than a politician".
His foreign minister, asked on a visit to NY to attend the U.N. General Assembly what the countermeasure would be, said his country may test a hydrogen bomb in the Pacific Ocean.
Mr. Trump's latest steps to punish foreign companies that deal with North Korea - in the form of further economic sanctions announced Thursday - was the most recent salvo in a US -led campaign to isolate and impoverish the government of Kim Jong Un until it halts its missile and nuclear tests.
Kim had delivered a tongue-lashing of his own - vowing to "surely and definitely tame the mentally deranged USA dotard with fire", in an address read out on state television by a star news anchor before a still image of Kim at his desk.
The Ministry of Commerce said in a statement on its website that China would limit exports of refined petroleum products from October 1 and ban exports of condensates and liquefied natural gas immediately to comply with the latest United Nations sanctions.
"DPRK is a responsible nuclear weapons state", he explained.
South Korea's Unification Ministry spokesman Baik Tae-hyun told reporters Friday that North Korea must immediately stop such provocations and return to talks on nuclear disarmament.
The tougher stance follows North Korea's latest nuclear test this month.
Experts say this move could cut off a major source of foreign currency for Pyongyang, as textiles are one of country's major exports, estimated by IHS Markit analysts to value $750m. The United States' only test of an operational ballistic missile with a live warhead was sacked from a submarine far out in the Pacific Ocean in 1962. "It is truly terrifying if something goes wrong".
The natural disaster, which South Korea put at magnitude 3.0, was detected in Kilju county in North Hamgyong Province, where North Korea's known Punggyeri nuclear site is located, the official said. The data comes after the United Nations Security Council unanimously voted on August 6 to impose a ban on exports of coal, iron, iron ore, lead, lead ore and seafood as punishment for intercontinental ballistic missile tests in July.
In recent months, North Korea has launched a pair of intercontinental missiles believed capable of striking the continental US and another pair that soared over Japanese territory.
Kim lashed out after Trump stunned the world on Tuesday when he used his maiden UN General Assembly speech to warn the North that Washington would "totally destroy" it if the U.S. or its allies were attacked. He also said he would "tame the mentally deranged USA dotard with fire".
Also Thursday, Trump issued an executive order expanding the Treasury Department's ability to target anyone conducting significant trade in goods, services or technology with North Korea and to ban them from the USA financial system.
Earlier in September, North Korea conducted its sixth and largest nuclear weapons test, and it has regularly conducted missile tests as rhetoric between it and the USA has continued to rise.
Though the majority of North Korea's imports come from China, Treasury Secretary Steve Mnuchin said "This action is directed at everyone" and the steps are "in no way specifically directed at China".
Apartment buildings in Pyongyang, North Korea, on September 22, 2017.
Russian Federation and China have both appealed for an end to the escalating rhetoric between Washington and Pyongyang, and Moscow's Foreign Minister Sergei Lavrov complained that that scrap resembled a "kindergarten fight between children".
Earlier this month the U.N. Security Council passed a new resolution with sanctions on North Korea.