Remove scales, gills, and guts from the fish. Rinse and cut crosswise into 3 steak pieces. Pat dry.
Arrange eggplant, onion, and ginger in the bottom of a 1 1/2-quart pot. Place fish on top. Add green chiles and sprinkle with salt.
Pour water and vinegar into the pot with the fish. Bring to a boil. Boil for 5 minutes; reduce heat and simmer until fish flakes easily with a fork, about 20 minutes. Add green bell pepper and bitter melon. Cook until bitter melon is soft to the touch, about 10 minutes more.