The North had named Guam among a list of possible targets for attack that included Hawaii and the US mainland.
North Korea is not thought to have the technology to strike the US mainland with either a nuclear weapon or a ballistic missile, analysts say.
The Kaesong complex is staffed mainly by North Koreans but funded and managed by South Korean firms.
Pyongyang blocked access for a second day on Thursday, and threatened to shut down the zone.
North Korea has issued an array of statements in recent weeks threatening nuclear strikes and attacks on specific targets in the US and South Korea.
It has announced a formal declaration of war on the South, and pledged to reopen a mothballed nuclear reactor in defiance of UN Security Council resolutions.
In its latest statement, attributed to a military spokesman, the North appeared to refer to ongoing military exercises between the US and South Korea in which the US has flown nuclear-capable bombers over the South.
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