"I like a watermelon" is proper English, but saying it like that means you like a particular watermelon. If we were talking I'd want to ask why you like "a" watermelon and not watermelon in general. If you like the fruit it would be more proper to say "I like watermelon." If you only liked a particular watermelon it would be less confusing to say "I like this watermelon".
If you would like to have watermelon you would say "I would like a watermelon", "I would like some watermelon", or "I would like some watermelons".
The main difference is the first sentence requests one watermelon, most people would assume you want one whole watermelon fruit. If you said the third sentence, people would assume you want two whole watermelons.
If you said the second, most would assume you're asking for part of a watermelon or part of multiple watermelons, like some watermelon cubes or wedges.