"Er isst seinen Apfel nicht."
"seine" would work for the plural "Äpfel" or a feminine word like "die Orange", but "Apfel" is masculine singular.
Can someone explain why "He won't eat his apple" is incorrect? It may have a slightly different meaning but feels a lot more natural than "He doesn't eat his apple".
There's no difference between Present Simple and Present Continuous in German like it's in English, so both "He doesn't eat his apple" (I realize this one sounds unnatural in English, so just for example) and "He is not eating his apple" are "Er isst seinen Apfel nicht" in German. This is exactly why you're allowed to use both of those translations.
Generally "He won't eat his apple" would be something like "Er wird seinen Apfel nicht essen", however difference between Present and Future tenses are not that stale in German, so in fact you could say "Er isst seinen Apfel nicht" to indicate the future. I think they could mark "He won't eat his apple" as an improper answer to not mess in our heads when we start covering real future tense. :)