Mr Trump responded that the “madman… will be tested like never before”.
The two countries have engaged in ever more heated rhetoric in recent months.
China responded to the war of words, warning that the situation was “complicated and sensitive”.
“All relevant parties should exercise restraint instead of provoking each other,” said Foreign Minister spokesman Lu Kang.
Russia also urged restraint. Kremlin spokesman Dmitry Peskov said Moscow was “deeply concerned by an escalation of tensions”
North Korea has been testing missiles at an unprecedented rate, and conducted its sixth nuclear test despite international condemnation.
North Korean Foreign Minister Ri Yong-ho, who had earlier compared Mr Trump’s speech to “the sound of a barking dog”, has warned that Pyongyang could test a hydrogen bomb in the Pacific Ocean in response to the US president’s threat.
“It could be the most powerful detonation of an H-bomb in the Pacific,” Mr Ri said, quoted by South Korea’s Yonhap news agency.
However, he added: “We have no idea about what actions could be taken as it will be ordered by leader Kim Jong-un.”
Mr Kim said in an English statement carried by state news agency KCNA that Mr Trump’s remarks “have convinced me, rather than frightening or stopping me, that the path I chose is correct and that it is the one I have to follow to the last”.
He said “now that Trump has denied the existence of and insulted me and my country in front of the eyes of the world and made the most ferocious declaration of a war in history”, North Korea would consider the “highest level of hard-line countermeasure” to make Mr Trump “pay dearly for his speech”.
He ended by saying he would “surely and definitely tame the mentally deranged US dotard with fire”.
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