The new order allows the U.S.to blacklist people and companies that conduct any business with North Korea, including shipping and trade.
Kim's statement responded to Trump's combative speech days earlier.
"North Korea has no intention whatsoever, of abandoning its nuclear or missile development", Abe said.
During that meeting, Moon praised Trump's speech Thursday as "strong", and said he believes it will "help contain North Korea".
North Korea has been testing missiles at an unprecedented rate, and conducted its sixth nuclear test despite worldwide condemnation. "Tolerance for this disgraceful practice must end now".
Trump has become known for using language similar to that usually employed by North Korea.
"Do business with the United States. or the lawless regime" of North Korea, he said.
"All our efforts are to prevent war and maintain peace", Moon told the United Nations in stark contrast to his US counterpart, and called for "peaceful, diplomatic and political solutions".
Trump's announcement came as other major players on the world stage announced similar measures to press North Korea. A top North Korean general called Trump's "fire and fury" threats "a load of nonsense" let out by "a guy bereft of reason".
Last week, North Korea threatened to "sink" the Japanese islands, saying Japan "no longer needed to exist near us".
North Korean leader Kim Jong-Un visits a fruit farm in South Hwanghae Province, North Korea in an undated photograph released on September 21, 2017.
Trump made the announcement during a lunchtime meeting in NY with leaders of South Korea and Japan - two USA allies that have been shaken by North Korea's nuclear saber rattling.
Trump told the U.N. General Assembly this week that the US will have no choice but to "totally destroy" North Korea if the North continues to threaten the USA and its allies, including Asian neighbors South Korea and Japan.
CBS News has also learned Trump is leaning toward decertifying the Iran nuclear deal.
South Korean President Moon Jae-in said Thursday he does not seek a North Korea regime collapse, during the 72nd General Debate at the United Nations General Assembly in NY.
The three leaders will also meet over lunch.