SEOUL—A South Korean transgender soldier faces a potential discharge this week after undergoing gender-reassignment surgery late last year, in a test case for the role LGBT people play in the socially conservative country.

Advocacy groups say it is the first incident of its kind in Asia. The soldier, currently a staff sergeant in her 20s, voluntarily enlisted as a male, but now wants to continue serving as a female after undergoing gender-reassignment surgery in November in Thailand. A military panel is expected to rule on...