The camera could be set or clipped in a position to record. It doesn't have to only be held. The view may not always be optimal, but the procedure would be recorded.
You would have to FIND a place to clip the camera, bring appropriate mounting hardware and hope it didn't get in the way of the doctor or that the doctor didn't get in the way of the view of the camera.
My father in law has a video set-up in his office for recording surgeries and it is NOT a simple apparatus. Often you need two or more cameras synced to get a half decent recording. Usually one person/assistant is completely dedicated to the recording. If it is a complex procedure, two people are used for making the recording.
Some GoPro cameras can be positioned on the head. It depends what the doctor has on their head if this filming view would work or not.
I have seen oral surgeries recorded by people wearing a GoPro on their head. Those videos are UNWATCHABLE, Every time the person moves their head, even a little, the screen jumps around, and since surgeons are constantly changing position to see what they need to see, the GoPro is not used by anyone wanting to make a decent video. Furthermore, the lighting exposure on the surgical field changes every time they move their head, so the screen goes light and dark as they move. There are companies who make head borne surgical cameras for recording procedures, but they are very rarely used because of the head movement problem and because the surgeon's telescopes frequently have a headlight so they can see better already attached.
I SPEAK FROM EXPERIENCE, NOT GUESTIMATES.
This is not as simple as it might appear if you want a good recording.
I am not saying you can't make a recording. If your standards are very low, you could make a bad recording that would show you were "in the room" when the procedure was done, but the quality would be low.