You can use either doctor appointment or doctor's appointment. Merriam-Webster indirectly supports the latter with its second example under its listing for appointment.
I agree with tchrist that the word doctor in doctor appointment is a noun (attributive noun I believe), although the definition of an attributive noun is a noun used as an adjective, so maybe it's just semantics at some point.
To the point above about the appointment being reciprocal, I can sort of get on board with that. I suppose the doctor does have an appointment with the patient as well, but I think the emphasis is a bit off in that perspective because I think the ownership pertains more directly to the patient (just my take).
In any case, you should be okay with either usage above.