leader, President Trump sounded an optimistic note Friday about future negotiations, saying it is even possible that a meeting could take place on June 12 as originally planned.
In a personal letter to Kim, Trump announced Thursday he would not go ahead with the June 12 summit in Singapore, following what the White House called a "trail of broken promises" by the North.
The move may have shocked the world but as the BBC's Heather Chen and Minji Lee of BBC News Korean write, it's an even bigger setback for the people at the heart of it: North and South Koreans. "This missed opportunity is truly a sad moment in history".
Christopher Hill, former USA ambassador to South Korea told CNBC, "Such a gap in expectations on both sides could have been avoided if the U.S. had gotten a better reading of North Korea's intentions, especially since Secretary of State Mike Pompeo had met with North Korean leader Kim Jong Un twice in the past few months".
"Denuclearization of the Korean Peninsula and the establishment of permanent peace are historic tasks that can neither be abandoned nor delayed", Yonhap quotes Moon as saying.
A statement by Vice Foreign Minister Kim Kye Gwan Kim said Pyongyang's "objective and resolve to do our best for the sake of peace and stability of the Korean Peninsula and all humankind remain unchanged".
In a separate statement at the White House, Mr Trump said the step was a "tremendous setback for North Korea and the world", adding the USA military was "ready if necessary" to respond to any "reckless" act from North Korea.
"But we shouldn't spend all our time thinking about the deal, without focusing on here's what this needs to look like in order to what president has said, to keep Americans safe from the threat that North Korea presents today", Pompeo added.
Trump said North Korea's recent "tremendous anger and open hostility" toward the USA made the summit inappropriate.
Other South Koreans had concerns closer to home.
Mr Putin said of Mr Trump's announcement, "in Russian Federation we took this news with regret".
The mother of twin boys fled North Korea in 2006.
"We are in a boxer stance, we are ready to respond", Lieutenant General Kenneth McKenzie, director of the United States military's Joint Staff, told reporters.
General Vincent Brooks, the commander of US Forces in South Korea, said: "The opportunity is not lost". We're talking to them [North Korea] now.
Decades of diplomatic stalemate had given way to what appeared to be genuine progress in one of geopolitics' stickiest issues, thanks, supporters said, to Trump's unorthodox approach and willingness to engage an global pariah.
Trump, in his letter calling off the summit, said it was canceled "to the detriment of the world".
"Trump has no interest in peace in our country. That's the right way toward the final resolutions of abduction, nuclear and missile issues", he said. "For me, it feels like North Korea is more of a normal country, saying it would give the USA time and wait", said Yun Hae-ri, a 25-year-old office worker.
References to Libya have angered North Korea.
A senior White House official said Pyongyang had demonstrated a "profound lack of good faith" in the run-up to the summit - including standing up the White House's deputy chief of staff, who had travelled to Singapore for preparatory talks.
Meanwhile, commenting on what lies ahead, Dr Sojin Lim, a senior lecturer in Korean studies told The Independent that "It will now be up to South Korea to try to bring the two countries back into some sort of dialogue - however, North Korea's burgeoning relationship with China means that it is now far less dependent on the United States, so it will be even more hard to re-establish negotiations between the two countries". The US balked at that demand.
"They very much want to do it, we'd like to do it", he said.
A spokesperson for Moon's office said on Thursday that South Korea was trying to understand the reason for Trump's abrupt cancellation.
President Donald Trump at the White House earlier this month.
Moon convened an emergency meeting with top security officials after Trump's cancellation.
South Korea has been left reeling from the U.S. president's announcement.
Typically, these talks end with North Korea walking away or taking provocative actions that force the United States to walk away.
Both Pence and Trump's hawkish national security adviser, John Bolton, had recently raised the spectre of Libyan leader Muammar Gaddafi, who gave up atomic weapons only to die years later at the hands of US-backed rebels. Pence said both the Clinton and Bush administrations had been "played" by the North Korean government.