Kim says 'deranged' Trump shows need for nuclear programme
The US president said on Tuesday that if America was forced to defend itself it would "totally destroy" North Korea.
Mr Trump also mockingly called Mr Kim a "rocket man" on a "suicide mission".
The two countries have been engaging in increasingly heated rhetoric in recent months.
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".
Experts say this is the first time a North Korean leader has made a direct address to an international audience.
The statement came shortly after the country's delegation arrived in New York for the UN General Assembly. Analysts say that for this reason it merits serious and thorough consideration.