"The boy likes seeing airports, not hospitals."
Translation:A fiú repülőtereket szeret látni, nem kórházakat.
The sentence is trying to tell us what the boy likes to see: airports, not hospitals. "Airports" needs to come immediately before the verb in order to convey that focus (and the "this not that" contrast that's intended). Your version places focus in the first part of the sentence on the boy instead. So the second part doesn't seem to follow.
If you spoke the English sentence aloud, you would use stress like this: "The boy likes to see AIRPORTS, not HOSPITALS." (Imagine yourself saying it sort of angrily to somebody who's misguidedly trying to take the boy to see a hospital.)
Your word order in Hungarian would be the equivalent of saying "The BOY likes to see airports, not HOSPITALS."
If you start a sentence with A fiú szeret repülőtereket látni, nem... then I am expecting to hear you name somebody else, contrasting with the boy. For example, A fiú szeret repülőtereket látni, nem a lány.
It should be noted that "proper" English also should use the infinitive here: The boy likes to see airports, not hospitals and that actually the English sentence here is not the best, since I'm guessing that the "stating a general preference of the boy" is what's meant.
Using the gerund seeing, while acceptable grammatically, changes the meaning more to: The boy, who is taking a tour of hospitals and airports, likes seeing the airports (which he is seeing on the tour) and does not like seeing the hospitals (which he is seeing on the tour). This is because seeing is a noun, which is a concrete thing the boy likes (and not the action of seeing, which is the infinitive to see)
Of course this is getting way into grammar pedantry because a lot of people aren't going to care much about the difference (except when it sounds weird).