1. If you like, I could go with you to the doctor’s.
2. If you like, I could go with you to the doctor.
[1] is correct. [2] isn't, but this sort of usage is common in spoken English - especially American.
Compare:
3. I'm going to the hairdresser this afternoon.
4. I'm going to the hairdresser's this afternoon.
[4] is correct, but many people wou