The South's national security adviser said on Friday that Mr Kim sent a letter to South Korean President Moon Jae-in saying the incident should not have happened and that the man was shot as part of measures to battle the coronavirus. But North Korea denied burning his body, saying it could not find him after the shooting.
The South Korean government employee went missing Monday from his boat near Yeonpyeong Island, about 10 kilometers (6 miles) south of the nautical border known as Northern Limit Line.
South Korean military officials say the man was interrogated while in the water over several hours and allegedly expressed a desire to defect, but noted that he was killed after an "order from superior authority".
According to the North Korean message, North Korean troops first fired blanks after the man did not fully explain why he was there, rather than saying he is from South Korea.
Defence Minister Suh Wook said it was believed the official was trying to defect, citing the man leaving behind his shoes on the fisheries ship and putting on a life jacket and resorting to a floating object when he was found in North Korean waters.
"Our military strongly demands North Korea provide explanations and punish those who are responsible", South Korean General Ahn Young Ho said at a press briefing.
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North Korean leader Kim Jong Un sent a congratulatory message to President Jokowi to mark Indonesia's 75th anniversary on August 17 this year. South Korean coast guard said earlier on Friday that their ships were searching waters near the boundary in case the official's body drifts back. The soldiers then approached the floating device, but only found "a large amount of blood", but not the man, according to the North's statement.
According to the message read by Suh, the troops determined he was dead and burned the floating object in line with anti-coronavirus rules.
Yang Moo-jin, a professor at the University of North Korean Studies in Seoul, said the apology showed Kim did not want the incident to escalate but it remained highly contentious in the South.
Seoul had condemned its rival over the incident on Thursday. Moon was said to have sent a letter to Kim on September 8, to which Kim responded four days later.
The military announced the fishery official's death a day after Moon proposed a new regional disease control and health initiative including North Korea to cope with crises like the coronavirus and strained ties with Pyongyang.
"In an unusual move, the North sent a letter under the name of Kim Jong-un and offered an apology, but our military has not even contacted (the bereaved family)", he told Yonhap News Agency.