Pyongyang's state-controlled network KCTV aired images of the launch on Wednesday and reported Kim attended the event.
North Korea is believed to have launched the Hwasong-12, an intermediate-range ballistic missile, previously tested in April and May.
The photographs of the missile on a road mobile launcher show the weapon both inside and outside North Korean facilities.
Images also showed the missile during launch, according to South Korean news agency Yonhap.
KCTV's images depict the Hwasong-12 missile rising into the atmosphere at several angles and disappearing from the screen, after the object emitted a flash of light.
North Korea stated the test was a demonstration against U.S.-South Korea joint exercises, the annual Ulchi Freedom Guardian, scheduled to conclude Thursday.
"In order to determine the operational capability of the medium-range ballistic rocket, a test of maneuvering anywhere and striking was carried out," KCNA stated.
KCNA also reported the North Korean leader will "keep an eye on U.S. actions" following the launch of the Hwasong-12.
Kim had made a similar statement following threats to launch a siege of Guam in August. The missile launch on Tuesday is believed to be a follow-up response to that statement, following the U.S. and South Korea decision to go ahead with joint drills, according to News 1.
But the North Korean leader's decision to "keep an eye" on U.S. responses is a sign he may be signaling a willingness to talk, according to the report.
Early Wednesday, U.S. President Donald Trump tweeted talks are not the answer.
"The U.S. has been talking to North Korea, and paying them extortion money, for 25 years. Talking is not the answer!" Trump said.
Trump had also said earlier "All options are on the table" following North Korea's missile launch.
The U.S. has been talking to North Korea, and paying them extortion money, for 25 years. Talking is not the answer!- Donald J. Trump (@realDonaldTrump) August 30, 2017