The Choe Hyon is one of two 5,000-ton destroyers in the North's arsenal, both launched this year as Kim seeks to ramp up the country's naval capabilities.
The warship "is a clear demonstration of the development of... (North Korea's) armed forces," Kim said during his visit on Sunday, according to state news agency KCNA.
"The tremendous ability of our navy should be exercised in the vast ocean to thoroughly deter or counter and punish the enemy's provocations for the sovereignty of the state," he added.
Kim has vowed to build a third destroyer of a similar class by October next year.
South Korea's military has said the Choe Hyon could have been developed with Russian help, possibly in exchange for deploying thousands of North Korean troops to support Moscow's war in Ukraine.
KCNA photos showed Kim seemingly overseeing a control room inside the ship with monitors showing the ocean around the Korean Peninsula.
Another photo showed the leader pointing to a blurred map in front of military generals.
Kim's visit came a day after he announced the deployment of "special assets to major targets" in response to what he called the United States' arms build-up in South Korea, without providing further details.
Pyongyang's enemies "will have to worry about which direction their security environment is moving in", he said on Saturday.
The United States stations about 28,500 troops in the South to fend off military threats from the nuclear-armed North, and last month conducted a joint military exercise with its security allies, South Korea and Japan.
Pyongyang routinely denounces such drills as rehearsals for invasion, while the allies insist they are defensive in nature.
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