I would have naively guessed you need a 'gehen' in there somewhere. Is there a rule here or is this just idiomatic?
As Christian said, the infinitive can often be dropped when it is implied. Same as in English: I would like a beer (to drink).
The meaning is very similar but I guess "zum" is the reason you should assume the intended verb is "Go". ZUM in this sentence means "To the" so you cannot say: I do not want to see to the doctor.
Yes it should be accepted, but I think it is by now?
@Natalino4ka: You're right, that "zum" includes a definite article and hence would have to be translated using "the" instead of "a". However, in THIS case, if someone says "Ich gehe zum Arzt", it doesn't necessarily mean they're going to a particular doctor - it just means they're going to a doctor. It's a common phrase. Therefore, in this case, "I don't want to see a doctor" is a perfectly fine translation.
I am not native speaker, but I think the article should be "the" not "a" because of "zum = zu + dem"
That's true when literally translated, but dativ nouns always require an article, and that article doesn't always translate directly back into English
I have problem with the place of nicht, why isn't it "Ich will zum Arzt nicht" ?
I'm not a native German speaker, but -- most often the verb goes after the finite verb: "ich will nicht zum..." meaning "I don't want to go..."
You can move the "nicht" to change the emphasis, and make it clear that the negation is about some other element in the sentence, but that just doesn't work here -- "will nicht" is really a single idea, and moving "nicht" away from "will" would just weaken the sentence and make it sound funny.
There's some good information about German word order in the following link -- search down for "placement of nicht"