Usage of articles in English sometimes seems totally haphazard or random. Many times, it is just an idiomatic usage.
As a native speaker, I can say that the following are idiomatic in American English:
Do you need to go to the hospital?
This is the case, even when we do not have a particular hospital in mind. And it doesn't matter whether there is one or more than one hospital that one can practically 'go to' (that is, is in the area). We use this for other typical locations, such as the library, the park, the grocery store, the mall, etc., even when there are many in a city or geographic area.
We also use the elevator even when there are five elevators one can take, and we do not have a particular elevator in mind. For example: 'Take the elevator (= any elevator) to the tenth floor' (not an elevator, although it would be grammatically correct).
The same for
Do you need to go to the doctor?
This works the same as the hospital, as no particular doctor is in mind. It could be that a native speaker is conceiving of the doctor as a location (similar to the doctor's), but I'm not sure; and I'm a native speaker. I just use the language as other native speakers do. I didn't invent idiomatic expressions; I inherited them.
Nevertheless, grammatically it is okay to use a in either sentence above.
And the following would only be said with the indefinite article, unless a particular doctor is in mind:
Do you need to see a doctor?
And the idiomatic expression:
Is there a doctor in the house?
The use of the definite article for certain buildings (the hospital) or places (the park) even when there are more than one in a given geographic locale, such as a city or town, is shown in the following joke:
Can you tell me how to get to the hospital?
No particular hospital is being referenced, just any hospital (which one would think by logic that American English speakers would use only the indefinite article for, but this is not the case). And the answer to the joke is
Yes, just go stand in the middle of the road over there and wait.
If you stand in the middle of the road, you will, sooner or later, get hit by a car, and you will get injured, and you will wind up at the hospital.