He added that United States continues to seek a "complete denuclearization of North Korea".
The president said "tolerance for this disgraceful practice must end now".
In a tirade against Trump responding to the U.S. president's thundering speech at the United Nations this week, the North's leader Kim Jong-Un warned of the "highest level of hard-line countermeasure in history".
President Donald Trump on Thursday announced new financial sanctions targeting North Korea as his administration seeks to build global support for more aggressively confronting the rogue nation, whose escalating nuclear and ballistic missile capabilities have reached what US officials consider a crisis point.
South Korean TV footage also shows Mr Ri saying he feels "sorry for his aides" when he was asked about Mr Trump's "Rocket man" comments.
"North Korea has no intention whatsoever, of abandoning its nuclear or missile development", Abe said.
Media captionHow would war with North Korea unfold? It wasn't clear what Trump was referring to. "I'd like to advise Trump to exercise prudence in selecting words and to be considerate of whom he speaks to when making a speech in front of the world", Kim said.
The president also said that the USA and South Korea were working on improving trade agreements though that was less important than addressing the North Korea threat.
President Donald Trump's announcement that Beijing told its banks to stop dealing with North Korea is "not consistent with the facts", a Chinese foreign ministry spokesman said Friday, but he gave no indication what steps China might be taking.
Trump spoke Thursday at a meeting in NY with the president of Afghanistan.