Park Wang-ja was killed on July 11 when she wandered into a military zone near the resort, one of only two areas north of the demilitairised which South Korean tourists are allowed to visit.
North Korea said the woman strayed into the restricted military area while strolling on a beach before dawn and refused to comply with a soldier's order to halt, instead running away before being shot twice.
South Korea suspended tourism to the resort after the incident, but Seoul's Yonhap news agency says there are at least 300 South Koreans still working at Mount Kunming.
'Military counter-actions'
Pyongyang also warned on Sunday that it carry out possible military action to any move against North Korea in the resort or in its military areas.
"We will take strong military counter-actions against even the slightest hostile actions in the tourist resort," the statement said.
"The passage of people and vehicles to Mount Kumgang through the military demarcation line will be more strictly limited."
The warnings came two days after Seoul cast fresh doubt on Pyongyang's account of the shooting.
A North Korean government spokesman said that the army was taking steps to cope with "grave provocation" from South Korea that is trying to pass on the blame on to the North.
Kim Ho-nyeon, Seoul's unification ministry spokesman, expressed regret at the move and renewed the demand that Pyongyang comply with its investigation of the July 11 shooting.
The two countries have technically remained in conflict since the Korean War ended in 1953, although relations have warmed somewhat in the last eight years.
The Korean conflict ended in a truce, but no formal peace treaty was ever signed.