'Not on my watch' means 'not while I'm still in charge' or, if referring to the past, 'it didn't happen while I was in charge.'
Example:-
--“No one died that night, not on my watch …. for which I've forever been eternally grateful.”:inno:
It's a good thing you mention it, since I've seen this translated as "όχι σύμφωνα με το δικό μου ρολόι", which is another phrase, really ("not by my watch").
In this case, I believe, "watch" is used in the sense of "time when one is on duty and, as a result, responsible for what happens." In other words, "όχι όσο κάνω εγώ κουμάντο" or "όχι στη βάρδια μου" or "όχι όσο είμαι υπεύθυνος εγώ", κ.ο.κ.