The 5000-tonne warship was equipped with the "most powerful weapons" and built "within 400-odd days perfectly with our own strength and technology", the KCNA report said, quoting Jo Chun Ryong, a secretary in the ruling Workers' Party.
Kim, in a speech from the launch reported by KCNA, said the warship would be handed over to the navy and go into service early next year.
Footage from North Korea's state-run television KRT showed Kim arriving in Nampho with his daughter Ju Ae after travelling to the western port city by train.
"If the US continues to renew its record in the protests of military power, we will have no choice but to renew it in the exercise of strategic deterrence," the leader was heard saying on TV footage.
Kim Yo Jong, the powerful sister of Kim Jong Un, also appeared in the TV footage.
The launch, which took place on Friday at the military shipbuilding dockyard of Nampho, marked a new era of the "great Kim Jong Un-style fleet building", KCNA said, quoting Vice-Admiral Pak Kwang Sop.
The ship was graded as the "Choe Hyon-class", named after anti-Japanese revolutionary fighter Choe Hyon, the report said.
Earlier this month, Reuters reported that North Korea's new class of warship was able to accommodate dozens of vertical launch cells to carry missiles its military has already developed, citing analysis of a satellite image.
Kim also said strong pre-emptive attack capabilities were the most "convincing war deterrent" and that there was no limit to the scope of such attacks.
"The security environment of our country is very serious right now," the leader was quoted as saying by KCNA while vowing to build a fleet for open-sea operations.
He also thanked the workers and technicians for building the new destroyer, which was "true to the party's line of strengthening the naval forces", according to KCNA.
Kim also said that the United States was deploying strategic assets in the Korean Peninsula on a regular basis.
The B-1B strategic bomber was used in a joint military drill between South Korea and the US earlier this month alongside fighter jets, according to Seoul's defence ministry.
B-1B bombers have featured regularly in joint military exercises in recent years.
North Korea condemns the drills as a rehearsal for war. Seoul says they are purely defensive.
14°C