Are U.S. military personnel stationed in Korea required to K-ETA application?

no. US Forces Korea (active duty) do not need to K-ETA application in accordance with facility and regional agreements under Article 4 of the ROK-US Agreement.Mutual Defense Treaty.Status of US Forces in Korea (SOFA).
However, you must present your ID (not your passport) to the airline when issuing your ticket.

Check ETA requirements