"Do you love me?"
It's just understood. You don't have to omit it but it's fairly common - and now, this is positive feedback - if something is understood and shorter→it will be commonly used→it will be understood even more→...
In general, it's a common thing to drop singular first/second person objects - even indirect objects - when the conjugation allows it (i.e for "engem/téged" it has to be indefinite). "Látsz?" can very well mean "Can you see me?" and not just "Can you see (at all)?" and noone really minds it. "Adom" can mean "I'm giving it to you" with the right context - here, definite conjugation ("adom" instead of "adok") expresses "it" by the way.
Have you ever wondered about "tetszik" for expressing likes? Grammatically, it's like appeal - the thing that you like (subject) appeals to you (indirect object). So, when I say "Tetszik ez a póló." ("I like this t-shirt"), nothing really expresses who likes it - it's just understood that if I make a statement about it, it probably "Tetszik nekem". The same way, if I ask "Tetszik ez a póló?", it's probably not "nekem" since I won't ask myself whether I like that t-shirt - it will be "neked", the most common thing is to ask you.
The same way we know "Why don't we make a cake for him?" is most probably a suggestion and not asking for a reason not to make a cake. :) It's simply based on common usage. (And common usage is probably based on definite-indefinite distinction, which only leaves "me" and "us" as the only meaningful options. And then "me" is simply overwhelmingly more common than "us", statistically.)
Szeretsz? = do you love? , but what is a possible context you would ask someone "do you love?" Do you mean "are you able to love?" (In this case, "Képes vagy szeretni?" is closer in Hungarian. )
With different personal pronouns:
Szeretsz engem? Do you love me?
Szereted őt? Do you love him/her?
Szereted azt/ezt? Do you love it?
Szeretsz minket? Do you love us?
So, if the sentence is Szeretsz? (and an additional pronoun is missing) the possibilities are: Do you love? Do you love me? Do you love us?
Do you love? is not used alone, so the most likely interpretation is Do you love me?
You can test it: Ask a Hungarian person "Szeretsz?". Will they understand it as if you asked "Do you love me?" Yes, definitely they will understand it that way :)
But szeretsz also means "do you like?" I fully accept what you say in terms of a Hungarian understanding the question in the way that you have described and you would know that far better than me. I remain, however, with the possibility of alternative interpretations and this isn't an area where we want to get something dreadfully wrong for fairly obvious reasons. Wouldn't it be safer to ask "Szeretsz engem?" I mean, she might be looking at a cake or a kitten when szeretsz alone is said!