I know I am rather late with this answer, but in order to do surgery your Physician needs to have 'Anesthesia' stocked in their inventory or stocked in the operating table. Anesthesia is Lordleaf, Sagewort, Valoroot and Englewood.
First you have to stabilize the pain via the 'Apply Anesthesia' interaction by moving the marker into the golden
sweet-spot area (pop-up in the upper left hand corner), then you can operate.
My level 2 Physician has operated successfully on several of my heroes when they went to her and asked for surgery after being gored by a unicorn or boar and after being mauled by a bear
. She has also performed surgery on patients in the clinic at this level. Sometimes a hero may have to be operated on twice to be fully healed
.