Yeah, that’s not how most physician offices work. They will most likely tell you to try the local urgent care or refer you to the ER.
The issue is that the average doctor is so busy that he has his whole schedule filled by existing patients for at least three weeks out. His staff knows how to plan the schedule for the existing patients, but a new patient takes longer than those who have been in before since the doctor has to take her history, review previous work and tests, if any, and figure out what her problems are. So it could be months before there’s a spot in the schedule big enough to accommodate an unusually long appointment.