@engineer,
Yeah, it's never shown. It's just Paul Newman cross-examining the surgeon and getting him to finally admit that the patient threw up in her mask and aspirated, thereby having her oxygen cut off.
I'm pretty sure the only time we see the patient is really briefly, and she's in a hospital bed with tubes everywhere.
There are definitely parts with hyperbole (the hospital essentially acting like the Mafia and threatening Newman), but it's otherwise a decent portrayal of what a small-time torts lawyer with a big case is like.