In a letter, the US president said he had been “very much looking forward” to the 12 June meeting in Singapore but that it was now “inappropriate”.
Mr Trump added: “You talk about your nuclear capabilities, but ours are so massive and powerful that I pray to God they will never have to be used.”
However, the president did not close the door for good on the chance of talks and said “someday” he hoped to meet Mr Kim.
The three-paragraph letter ended: “If you change your mind having to do with this most important summit, please do not hesitate to call me or write.
“The world, and North Korea in particular, has lost a great opportunity for lasting peace and great prosperity and wealth.
“This missed opportunity is a truly sad moment in history.”
Mr Trump’s mention of “anger and open hostility” appeared to reference an overnight statement by North Korea’s vice-minister of foreign affairs, who was hitting back at comments made by US vice-president Mike Pence.
Choe Son Hui called Mr Pence “ignorant” and “stupid”, and again raised the prospect of a “nuclear to nuclear showdown”.
She added: “We can also make the US taste an appalling tragedy it has neither experienced nor even imagined up to now.”
That rhetoric was in retaliation for remarks Mr Pence made in an interview with Fox News over concerns the North was not committed to denuclearisation.
Mr Pence suggested the North might end up like Libya, whose ex-leader Muammar Gaddafi was killed in an uprising years after giving up atomic weapons, if a deal was not made.
President Trump, speaking to reporters after his letter was published, warned Mr Kim against any “foolish or reckless acts”.
“If and when Kim Jong Un chooses to engage in constructive dialogue and actions, I am waiting,” said the president.
“In the meantime, our very strong sanctions – by far the strongest sanctions ever imposed – and maximum pressure campaign will continue, as it has been continuing.”
There had been talk of Mr Trump getting the Nobel Peace Prize for his bridge-building – but such possibilities are now firmly on hold.
Plans for the unprecedented summit had come after a rapid warming of relations between North Korea and the US, with the secretive regime pledging to end its nuclear programme and releasing three US prisoners.
But, speaking to a senate committee on Thursday, secretary of state Mike Pompeo – who had met Mr Kim face to face in preparation for the summit – said America had repeatedly tried in recent days to connect with North Korea over logistics for the summit, but got no response.
The same day Mr Trump pulled the plug on talks, Sky News was the only British broadcaster to witness the destruction of what North Korea says is its nuclear weapons test site.
South Korea’s president, Moon Jae-in, who met Mr Kim recently at a historic peace summit on the countries’ border, summoned his advisers for crisis talks after hearing the summit had been cancelled.
“I am very perplexed and it is very regrettable that the North Korea-US summit will not be held on 12 June when it was scheduled to be held,” Mr Moon said, according to the Yonhap news agency.
More from Donald Trump
Donald Trump banned from blocking Twitter users from his official account
Sinkhole opens up in White House lawn – and social media is loving it
Donald Trump may extend UK visit to play golf
Donald Trump says summit with Kim Jong Un may not happen
Bill Gates: Trump did not know difference between HIV and HPV
Trump questions whether Kim wants North Korea summit – but US still preparing
United Nations Secretary General Antonio Guterres said he was “deeply concerned” that the historic meeting appeared to be in tatters and warned the two sides they would need “nerves of steel” to get the process back on track.
In the UK, a spokesman for the prime minister said: “We are disappointed that the meeting will no longer go ahead as planned. We need to see an agreement that can bring about the completely verifiable and irreversible denuclearisation of the Korean peninsula and we will continue to work with our partners to this end.”