Combine salmon, kale, green onions, and ginger in a saucepan; add water and turn heat to medium-high. Bring liquid to a boil; reduce heat and simmer until salmon flakes easily with a fork, about 10 minutes. Add more water if needed. Season salmon with salt and pepper.
Stir rice into salmon mixture; cook until heated through, 2 to 3 minutes. Remove saucepan from heat. Stir seaweed into mixture and top with sesame oil.