Ri said that Trump's bombast had made "our rockets' visit to the entire U.S. mainland inevitable" and linked it to Trump's insulting shorthand references to Kim.
Harsh sanctions placed on North Korea's trade with the outside world will have no impact on its ability to complete building a nuclear bomb capable of reaching the United States, Ri said, suggesting that stage is imminent.
"Through such a prolonged and arduous struggle, now we are finally only a few steps away from the final gate of completion of the state nuclear force," he said.
"It is only a forlorn hope to consider any chance that the DPRK would be shaken an inch or change its stance due to the harsher sanctions by the hostile forces," he said, using the acronym for North Korea's official name, the Democratic People's Republic of Korea.
In response to Ri's threats at the U.N., Trump tweeted Saturday night: "If he echoes thoughts of Little Rocket Man, they won't be around much longer!"
A mass rally was also held Saturday in central Pyongyang to express support for "final victory" over the United States, the regime's KCNA news agency reported. The rally in Kim Il Sung square was attended by over 100,000 people, KCNA reported, "full of the spirit of annihilating the enemies." Speaking at the rally, a Pyongyang official said the people of the capital were "aflame" with the desire to wipe their sworn enemy, the U.S. imperialists, off the globe.
The rhetoric between Trump and Kim has grown exceptionally personal. At a rally Friday night in Alabama, Trump called Kim "little rocketman," magnifying the disparaging label he slung at Kim in his U.N. speech Tuesday in which he threatened that the United States would "totally destroy" North Korea in defense of itself or its allies. He said Kim was on a "suicide mission."
Kim, in turn, called Trump a "frightened dog" and a "mentally deranged U.S. dotard." Ri echoed those sentiments Saturday, calling the president a "mentally deranged person full of megalomania" and at one point referring to him as "President Evil."
Ri emphasized that North Korea has the know-how to carry out its threat. He said Pyongyang has a hydrogen bomb that can fit on an intercontinental ballistic missile capable of reaching the United States.
On Friday, Ri said Korea was prepared to test a hydrogen bomb over the Pacific Ocean.
"Trump might not have been aware what is uttered from his mouth, but we will make sure that he bears consequences far beyond his words, far beyond the scope of what he can handle even if he is ready to do so," Ri said.
On Friday, Trump kept up his verbal fusillade against Kim, tweeting that Kim is a "madman" who will be "tested like never before."
The heated exchanges between Trump and North Korea's leader come as diplomatic pressure may be starting to bear fruit. Secretary of State Rex Tillerson said last week that sanctions are beginning to have an effect.
China, North Korea's economic lifeline, has gradually imposed greater economic sanctions on its neighbor, including caps on oil. On Thursday, Trump signed an executive order giving the Treasury Department more authority to cut off trade that helps finance North Korea's weapons and nuclear programs.
While Tillerson has insisted that diplomacy still has a chance to work, military force appears to be increasing as an option.
Just before Ri spoke at the U.N., the Pentagon disclosed that the U.S. Air Force B-1B Lancer bombers flew in international airspace east of North Korea. The Pentagon said it was the farthest point north of the Demilitarized Zone separating North and South Korea that any U.S. planes have flown in this century.
South Korea's presidential office said Seoul and Washington had coordinated closely over the deployment of the U.S. bombers, calling it one of the most effective countermeasures against the advancement of North Korea's nuclear and missile programs, South Korean media reported.
The bombers took off from an air base in Guam, which North Korea has threatened to target. They were escorted by F-15C fighter jets from Okinawa, Japan.
Simon Denyer in Beijing contributed to this story.