Melting isn't the issue, it's the frequency-dependent resistance presented by most speakers. Introduce an interplay between that and the cable and you'll alter the frequency-response (irrespective of volume). This is also why a thinner cable is more acceptable for a shorter run.
For reference, you should be considering amps (unit, not the component) for a conductor, not Watts. For example, if you're feeding 676W into a 4 ohm load (overkill, I realise), that's 52V, 13A. The same current as a conductor would experience at 240V and 3kW. Obviously, you should also consider the breakdown voltage of any insulators too since, on both counts, house fires aren't too neighbourly.