North Korean state TV, for its part, hailed Kim for "opening a new chapter" in relations with the US. There was no mention of a declaration of North Korea's nuclear assets, which normally precedes any arms-control negotiation, or of timelines or deadlines.
Even as Pompeo staunchly defended the summit results, he was less exuberant than Trump, who tweeted on his return to the USA on Monday morning: "Just landed - a long trip, but everybody can now feel much safer than the day I took office".
But at the summit, he said, Trump strengthened Kim's hold on power.
"No more rocket launches, nuclear testing or research!"
After almost five hours of unprecedented talks between Trump and Kim on Tuesday, the two leaders signed a document in which Kim "reaffirmed his firm and unwavering commitment to complete denuclearization of the Korean Peninsula" and the United States agreed to "provide security guarantees".
In a series of tweets, Trump sought to take political credit for the summit but risked undermining the U.S. strategy in the region.
The Pentagon said Tuesday it was consulting with the White House and others, but was silent on whether the August exercise would proceed.
South Korea's presidential office said it needs to seek clarity on Mr Trump's intentions are after he said Washington will stop joint military exercises. However, they've long wanted a meeting with a United States president.
Despite all the uncertainty and scepticism following the summit, the change of tone between Washington and Pyongyang is astonishing. And how can the U.S. make sure Kim will keep his commitments?
How can anyone be sure that North Korea is really denuclearizing?
Pyongyang has reason to feel confident after the meeting, where the leader of the world's most powerful democracy shook hands with the third generation of a dynastic dictatorship, standing as equals in front of their nations' flags.
The spectacle was a major coup for an isolated and heavily sanctioned regime that has long craved global legitimacy.
Trump's Tuesday meeting with Kim produced a wave of USA concessions in exchange for vague, nonbinding promises from North Korea, but it did make nuclear war all but inconceivable at the present moment.
Annual military drills between the US and South Korea have been a major source of tension on the Korean Peninsula.
Even worse, Trump surprised South Korea and even other USA officials when he appeared to announce a US concession that was never mentioned publicly when he was with Kim: the suspension of the U.S.
"I suppose we could argue semantics, but let me assure you it's in the document", Pompeo said.
The upcoming joint exercise in August is more computer-based training. USA defense analysts have said Kim retains as many as 60 nuclear bombs and a range of missiles, including some he says can strike the U.S. It called the summit "dumbfounding and nonsensical", and said it will allow North Korea to permanently maintain its nuclear weapons program.
"Following this historic summit, I remain skeptical but hopeful that this new dialogue can translate into meaningful progress", Portman said in a statement.
President Trump continued to pat himself on the back Tuesday night for arraigning the Singapore summit - saying it helped reduce the risk of "potential Nuclear catastrophe". "You know, human rights is a forgotten issue", said Phil Robertson, the deputy director of Human Rights Watch's Asia division. "But the substance needs to be followed up", Akira Kawasaki, from the ICAN anti-nuclear group.