Well technically if a fish doesn't stay on the move to keep water flowing through their gills they CAN drown.
That's true for sharks - they can't "gulp" water as well as freshwater fish (or most bony fish, for that matter) making them at risk of drowning if stuck in a fishing net.
However, a fish can drown if their gulping reflex is disrupted, thus stopping the flow of water passing over her gills, so you're correct about that aspect. Unconsciousness can do that.