The president, commenting as he left the White House for a commencement speech, said it was even possible the summit could take place on the originally planned June 12 date.
North Korea on Thursday "completely dismantled" its Punggye-ri nuclear test ground to "ensure the transparency of discontinuance of nuclear tests", after blowing up tunnels at the site, it said.
The tone from both sides was warmer on Friday.
"His sudden and unilateral announcement to cancel the summit is something unexpected to us and we cannot but feel great regret for it", Kim Kye Gwan said, while noting that North Korea remained open to resolving issues with Washington "regardless of ways, at any time".
"We believe as the parties directly engaged on the issue, the summit of the DPRK (North Korea is formally known as Democratic People's Republic of Korea) and United States can play a crucial role for promoting the denuclearisation of the peninsula", Lu said.
Pence and Trump's hawkish National Security Adviser John Bolton had angered Pyongyang by raising the spectre of Libyan leader Moamer Khadafi, who gave up atomic weapons only to die years later at the hands of US-backed rebels. "Only time (and talent) will tell!".
The president's surprise exit capped weeks of high-stakes brinkmanship between the two unpredictable leaders over nuclear negotiating terms for their unprecedented sit-down. "They even detonated their nuclear test site", said Eugene Lim, a 29-year-old office worker in Seoul. But it also followed escalating frustration - and newly antagonistic rhetoric - from North Korea over comments from Trump aides about USA expectations for the North's "denuclearization".
The White House has repeatedly offered mixed messages.
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.
The senior USA official said the North violated a pledge to allow worldwide inspectors to monitor the supposed implosion of the test site. CBS News correspondent Ben Tracy was the only US broadcast network correspondent to witness it, among a small group of global journalists.
NFL players mulling new anthem protests 'to spite' league over planned crackdown
Rooney said input from the military and veterans' communities was gathered as well. "Whatever we do, whatever happens here". Carroll said it was too soon to know how the team might react but that whatever the team does it will do together.
Harvey Weinstein expected to surrender on rape charges
Al Franken, chef Mario Batali , casino magnate Steve Wynn and, most recently, New York Attorney General Eric Schneiderman. The statute of limitations for rape in NY was eliminated in 2006, but not for attacks that happened prior to 2001.
Rusal CEO, board members quit amid United States sanctions pressure
Deripaska, in turn, announced he was ready to reduce its stake in the largest shareholder of Rusal, En+, below 50%. The US will consider lifting the sanctions if Deripaska relinquishes control of his companies.
Trump's announcement came a day after Pyongyang hardened its rhetoric, calling comments by Vice President Mike Pence "ignorant and stupid". "We have to get it right", he said.
On Friday, North Korea's vice foreign minister said his country's "objective and resolve to do our best for the sake of peace and stability of the Korean Peninsula and all humankind remain unchanged".
It seems that much like their South Korean counterparts, officials in the DPRK were as surprised by Trump's abrupt cancellation letter as anyone.
Earlier, North Korean Vice Foreign Minister Kim Kye-gwan said that despite Trump's decision to cancel the June 12 meeting in Singapore, Pyongyang has the "intent to sit with the United States side to solve the problem regardless of ways at any time".
Still, some analysts are anxious the canceling of the talks could prompt a resumption in hostilities on the Korean peninsula, including renewed shorter-range missile tests or stepped-up cyber attacks by Pyongyang and increased sanctions or deployment of new military assets by Washington.
Japanese Foreign Minister Taro Kono said Tokyo understands Trump's decision to cancel the Singapore summit, Kyodo News reported.
He looked past the warnings of senior aides when he accepted Mr Kim's invitation to meet back in March. He unveiled the date and the time with characteristic showmanship. But again, in the past, North Korea attacked others including Trump using worse language.
Though, it didn't sound like Trump was personally toiling away at summit preparations.
First Vice Foreign Minister Kim Kye Gwan called Trump's decision "unexpected" and "regrettable".
USA defense and intelligence officials have repeatedly assessed the North to be on the threshold of having the capability to strike anywhere in the continental US with a nuclear-tipped missile - a capacity that Trump and other USA officials have said they would not tolerate.
North Korea says that, unlike Libya, it is a fully fledged nuclear state.
He was not informed of the decision before Mr Trump's announcement, reports said. They never told us anything. Kim paid a secret visit to his primary ally just before Pompeo's visit, and China is wary of any shift in the balance of power on the Korean peninsula.