SEOUL, South Korea — North Korean leader Kim Jong Un condemned South Korean-U.S. military drills and vowed a rapid expansion of his nuclear forces to counter rivals, state media said Tuesday, as he inspected his most advanced warship being fitted with nuclear-capable systems.

Mr. Kim’s visit to the western port of Nampo on Monday came as the South Korean and U.S. militaries kicked off their annual large-scale summertime exercise to bolster readiness against growing North Korean threats. The 11-day Ulchi Freedom Shield, which the allies describe as defensive, will mobilize 21,000 troops, including 18,000 South Koreans, for computer-simulated command post operations and field training.

North Korea has long denounced the allies’ joint drills as invasion rehearsals and Mr. Kim has often used them to justify his own military displays and testing activities aimed at expanding his nuclear weapons program.

The Korean Peninsula remains in a technical state of war, divided by the Demilitarized Zone into North Korea and South Korea.

While inspecting the warship Choe Hyon, a 5,000-ton-class destroyer first unveiled in April, Mr. Kim said the allies’ joint military drills show hostility and their supposed “will to ignite a war,” the North’s Korean Central News Agency said. He claimed that the exercises have grown more provocative than before by incorporating a “nuclear element,” requiring the North to respond with “proactive and overwhelming” countermeasures.

“The security environment around the DPRK is getting more serious day by day and the prevailing situation requires us to make a radical and swift change in the existing military theory and practice and rapid expansion of nuclearization,” KCNA paraphrased Mr. Kim as saying, using the initials of North Korea’s formal name, the Democratic People’s Republic of Korea.

Kang Yu-jung, spokesperson for South Korea’s new liberal president, Lee Jae Myung, who wants to improve ties with the North, said Seoul has “always regarded the Ulchi exercises as defensive” but offered no further comment on Mr. Kim’s remarks. South Korea’s Defense Ministry said it had no immediate new assessments to share regarding the North Korean warship’s capabilities.

South Korean and U.S. military officials say Ulchi Freedom Shield will focus on countering North Korea’s advancing nuclear and missile threat and will include training to deter North Korean nuclear use and respond to its missile attacks. The exercise will also incorporate lessons from recent conflicts, including Russia’s war in Ukraine and the clash between Israel and Iran, and address threats from drones, GPS jamming and cyberattacks.

Mr. Kim has hailed the development of his naval destroyer, Choe Hyon, as a significant advancement toward his goal of expanding the operational range and preemptive strike capabilities of his nuclear-armed military. The North unveiled a second destroyer of the same class in May, but the vessel was damaged during a botched launching ceremony at the northeastern port of Chongjin. The North has said the new destroyer, named Kang Kon, was relaunched in June after repair.

During Monday’s visit to Nampo, Mr. Kim also reviewed North Korean efforts to complete a third destroyer by October, KCNA said.

Tensions on the Korean Peninsula have worsened in recent years as Mr. Kim accelerated his military nuclear program and deepened alignment with Moscow following Russia’s invasion of Ukraine.

In his latest message to Pyongyang on Friday, Mr. Lee, who took office in June, said he would seek to restore a 2018-inter-Korean military agreement designed to reduce border tensions and called for North Korea to respond to the South’s efforts to rebuild trust and revive talks.