The dictator's remarks were reported by state media KCNA on Monday, although it did not say if the vessel would carry nuclear weapons.
North Korea has accelerated efforts to modernise its navy as Kim pursues a broader expansion of the country's nuclear-capable forces.
At a commissioning ceremony in the eastern port city of Wonsan on Sunday, Kim said the new ship's deployment to the sea east of the Korean Peninsula sent a clear signal to North Korea's adversaries.
"The time has come for us to exercise our sovereignty at sea and underwater," Kim said, according to KCNA.
The 5000-tonne-class destroyer, the Kang Kon, is the second warship of its class to enter service, following the commissioning of the Choe Hyon destroyer on North Korea's west coast about three months earlier, KCNA cited Kim as saying.
Kim said North Korea's naval nuclear armaments were advancing, and the country would continue to expand what he called its most powerful strategic assets in key waters to protect its maritime sovereignty and regional security.
He added that the Kang Kon would form part of North Korea's state nuclear response system.
KCNA did not specify what weapons systems might be deployed on the Kang Kon, but said is has the same weapons systems as the Choe Hyon.
A previous KCNA report said Choe Hyon-class destroyers can carry a range of weapons, including missiles that analysts say are capable of carrying a nuclear payload.
Kim said he would demonstrate another stage of North Korea's naval buildup in eight months, KCNA said.
Kim also said the construction of naval bases on North Korea's east coast was underway and that the country would create new naval units and build different classes of warships.
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