"I like watermelons" refers to your preference. It is equivalent to "I like apples." You are making a statement to do with your likes and dislikes. "I would like a watermelon" is what you say to the shopkeeper when you intend to buy one. You are indicating what and how much of it you want. You want one only. "I would like some watermelons" means you want to buy a number of watermelons. "I would like some watermelon" is how you order in a restaurant when you want a slice or when you want your fruit platter or fruit salad to include some watermelon pieces. Here "watermelon" is used as an uncountable noun referring to the flesh of the fruit.

Even though it's grammatically correct, "I like a watermelon." doesn't sound natural. However, the other one does sound natural, "l would like some watermelons." If you are requesting a single watermelon then it should be, "l would like a watermelon." I personally will add please like, "l would like a watermelon please." or more than one, "l would like some watermelons please."

"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.

yes, they are both correct. Like and Would Like have different meanings. Would Like means Want.