As far as I know, you get an UCI number the first time you apply with CIC and it stays with you forever. So you should have one already, and you can find it on any document you received in the past from CIC.
I'm not sure how this will help you with the doctor though. Because there are two types on medical exams:
1. Up front (and this is only for spouse, common law and dependent applications) - where you go there with the passport and the doctor gives you either an eMedical receipt that you include in your application - if he works with the electronic system - , or a copy of the IMM1017B (Up front medical report form) - if he works with the paper system.
or
2. Wait for instructions once you have submitted your application. When you opt for this method, the visa office will send you a form IMM1017E that you must take to the doctor together with 4 recent photographs.
It doesn't say anywhere on the CIC website that a medical exam can be done using the UCI number, so I'm not sure if that's the best way to go. I wouldn't trust this doctor, you said it yourself too that he has no idea what he's talking about.