How does a hospital prove they didn't perform a tubal ligation? You're asking them to prove a negative, often an impossible task. Short of videotaping every operation I don't see how this is possible.
... and a lot of hospitals do videotape every operation as a matter of routine.
A lot of hospitals also have routine post-op paperwork detailing what was done, paperwork that gets signed off on by all participants. If a doctor did a tubal and didn't put it on the paperwork, that would raise a red flag with the nurse (who watched him do it). The nurse might even mention it in her own report.
If the hospital introduces the report signed off by the surgeon, the assisting surgeon, the nurse, and so forth, and it doesn't mention a tubal, that's pretty good evidence that no such thing happened.
But if the hospital can't find the report,... well, that damns them. The obvious implication, which I would not hesitate to draw, is that the hospital HAS the report, that it mentions doing an unauthorized procedure, and that they're suppressing it. And I have no problem in this case taking the woman's unsupported word against the silence of the hospital where they should have been able to speak definitively. This is one of those instances where "her word against theirs" is
not even, because the hospital shouldn't have to rely on just words.
And if the hospital doesn't keep reliable records of what operations are performed, they should lose the case (and their license) for lack of professionalism.