Sea water is not that conductive: about 5 S/m. By comparison, the conductivity of copper is about 60000000 S/m, so seven orders of magnitude more conductive.
The low conductivity of sea water will effectively appear as a resistance in series with the coil. This will increase losses in the coil, reducing sensitivity.
But worse, it will decrease the Q factor of the LC filter, reducing selectivity. The resulting "radio" would be incapable of tuning to an individual station. Perhaps this isn't too much of a concern, since given the sensitivity issues you will have to be very close to a transmitter to hear anything, at which point that single transmitter is probably the strongest source of RF around.
I guess technically we could call this a "radio", though by this definition you could attach just about anything conductive to a diode and call it a radio. For any practical application, copper (or really, most metals) is a better choice: it's smaller, cheaper, electrically superior, non-corrosive, and solid at room temperature.