"You are about to see something on the Fox News that has never happened, we believe, in the history of White House reporting", reporter Steve Doocy said live on air.
Trump, speaking to reporters outside the White House, said he had given his "very direct" telephone number to Kim, adding: "He can now call me if he has any difficulty".
A 2014 United Nations Human Rights Council report found that North Korea's human rights abuses include "extermination, murder, enslavement, torture, imprisonment, rape, forced abortions and other sexual violence, persecution on political, religious, racial and gender grounds, the forcible transfer of populations, the enforced disappearance of persons and the inhumane act of knowingly causing prolonged starvation". I want my people to do the same, Trump said.
South Korea was left reeling this week after Trump announced following his summit with Kim that the exercises, which have taken place routinely for years on the Korean Peninsula, would stop.
North Korea no longer poses nuclear threat - Donald Trump
The U.N. nuclear watchdog has not had access to North Korean nuclear facilities since its personnel were expelled in 2009. We're hopeful that we can achieve that in the two and half years", he said.
"That problem is largely solved".
The surprise appearance Friday on "Fox & Friends", followed by a combative round of questions with reporters, came days after Trump returned from the Singapore summit expecting a hero's welcome and tweeting that the world now could "sleep well".
Trump has been calling lawmakers to express enthusiasm for the agreement - but also complaining that he has not had more robust support from GOP lawmakers, said a person with knowledge of the calls, who spoke on condition of anonymity to share internal conversations.
"We get along very well, we had good chemistry", Trump said. "This was long before I got there", Trump said.
Trump said there would be a "very strong verification process" for North Korean denuclearization. So I know you were trying to give him a compliment, but that was really more of an insult.
Trump was met with criticism this week after footage surfaced of him saluting a North Korean general. "Don't let anyone think anything different".
Japanese Foreign Minister Taro Kono is considering talks with his North Korean counterpart on the sidelines of a meeting of ASEAN envoys in Singapore starting in late July, the Yomiuri Shimbun reported.
Recently, Park filmed a video op-ed imploring Trump to do the right thing when he meets Kim Jong Un.