It's certainly one of the keys. But the point is this: it costs money to put in artificial reefs. Who is going to put up that money when nobody owns the fish? If you're a fisherman, it's better to save your dollars and wait for another sap to come along and spend his money, and then scoop up the fish he grew. Better even to wait for the inexhaustible American taxpayer to put up the money.
But if you own the fish, well, then, your incentive for building reefs and pumping iron solution into the sea is clear.
When flying over S. Korea a few years ago I noticed "tracts" of sea marked by buoys and bound by nets. I don't know exactly what they were farming (sea weed? fish?) but it seemed clear that locals were making use of these parcels to farm a profit.
I can see ownership of parcels of sea in areas currently owned by the respective countries (no conflict there). Different parcels may be best suited for different types of farming, and would be subject to valuation via auction.
This doesn't deal with international waters, but does give incentive to negotiate or buy ever larger segments of ocean. Therefore the ownership would track with the ability to render it profitable in an incremental fashion.
In the early west we didn't parcel out ownership of the wildlife but rather the land itself. In time fences went up. Back then it was probably as inconceivable of fencing large tracts of land in the midwest as it seems inconceivable that we may someday fence off large tracts of sea and ocean. But maybe thats where we are headed. It would be ownership as you say, but ownership of sea not fish.