Surgery as a noun is also performance of a surgical procedure. Surgical is an adjective. I couldn't find anywhere a verb for surgery. Like how you 'operate' for the noun operation.
The verb is 'to operate'. Sense 4 of 'to operate' from the OED is specifically a surgical sense:
a. intr. To perform an operation. Also with on, upon.
["operate, v.". OED Online. September 2015. Oxford University Press. http://www.oed.com/view/Entry/131741 (accessed November 06, 2015).]
Notably, the OED Thesaurus lists no synonyms for this, the surgical sense of the verb 'to operate'.
Consider the medical specific surgerize.
The patient was subsequently seen by Reuben Washington, M.D. and was surgerized at least twice for a shoulder injury New York Court of Appeals
We can add to the dictionary! I was looking for this word too, and though didn't find it, believe this makes complete sense, especially with some of the etymology others have stated:
surgered -- past participle of v. surger, "Be careful of the surgered knee."
If Merriam-freaking-Webster can change the definition of literally to mean figuratively, we can damn well make a verb from a noun. Happens naturally all the time as language evolves.