I think this might be possible. I don't really know, but if you've ever "cycled" an aquarium, it's pretty interesting and makes me think it's plausible.
If you put fish in an aquarium full of fresh water, they will generate ammonia and eventually die (unless they are very hardy fish).
So to "cycle" the aquarium, you can put drops of ammonia in the aquarium each day. Over time, bacteria that metabolize ammonia will enter the aquarium water and colonize it. These bacteria will remove the ammonia, producing nitrites (which are also toxic to fish, but less so than ammonia).
If you continue cycling the aquarium, more bacteria colonize the water, ones which metabolize the nitrites into nitrates.
Fish are usually ok with nitrates, so at this point, you can introduce the fish to the aquarium, and they will survive because the system will naturally remove the ammonia and subsequent nitrites.
Optionally, you can add plants to the aquarium and they can utilize the nitrates and remove them. Remember nitrates? From fertilizer?
Long story short... I'll bet the ocean can do amazing things like this. Don't know for sure, but it's plausible.