I agree that my father (who is Italian, or more accurately, Sardinian) would only use melone but the word that we use for watermelon is something funny. Maybe it's Sardinian. (That info is hard to find on the web, I'll have to ask my dad.) By the way, there are a gazillion different types of melons in Sardinia, all seeming to be called melone. Sometimes you can find them at the stores (on the mainland) here under the heading "fancy melons".

By the way, I googled "Italian watermelon" and got "cocomero". ?!

Okay, I found the word - Anguria - which appears to be on a menu in a Sardinian restaurant in Texas. ?!?!?! NOW my head hurts!