I've been thinking about this question on and off for a while; even asking a frum surgeon I know for his thoughts. I didn't initially understand his answer, but now I think I do, based on my recent studies in the halachos of pikuach nefesh on Shabbos.
The rules for pikuach nefesh (saving of life) on Shabbos are that everything necessary must be done (Shemiras Shabbas K'hilchasa 32:1), even if there is only a concern (חשש) that there is present danger (ibid, 1, 18, 19), even if there is only a slim chance of survival (ibid, 20). We also do things if they make things easier for the patient (ibid, 18, 22; see also this MY question).
However, things that have nothing to do with saving the patient are not allowed to be done (SS"K 32:41, 47); and, at all times, things that are not needed either to increase quality of care or to speed things up when time is of the essence, are not allowed to be done, or should be done with a lower level of prohibition (e.g. a rabbinic prohibition instead of a biblical one, or one prohibition instead of two; see SS"K 32:27-30), or through a non-Jew.
Even things that don't directly heal, are allowed if they're needed or requested by the patient, but then only with rabbinic prohibitions and שינוי [where it's possible to do it with a שינוי] (ibid, 24-25)
Therefore, I would say that he should borrow a watch from someone, or go directly to the OR after davening.
If he can't, it depends:
- If setting the watch must be done and there is no time to do it in
any other way, the doctor is not only allowed to, but must do it himself.
- If it's not necessary (which I think is probably the most likely as per my discussion with the surgeon, and a comment on the question), then he's not allowed to do it at all.
- If it is needed, but it's not urgent (which, if it is needed, is the most likely case, IMO), then I would say to have a gentile do it; or, if no gentile is available, to do it in a different way than is usually done ("שינוי") [with his teeth?].