My first thought was that you might glimpse your watch:
See or perceive briefly or partially.
‘he glimpsed a figure standing in the shade’
(from the Oxford Dictionaries)
But then I felt that wasn't quite right. What you really did was catch sight of your watch:
catch (or get a) sight of
Glimpse for a moment; suddenly notice.
‘when she caught sight of him she smiled’
So in your sentence, it would be:
I caught sight of my watch... shoot! It's 10:30!
An anonymous user (thanks, whoever you are!) noticed that the Oxford Dictionaries' definition of glimpse as a noun offers yet another option via its sample sentence:
A momentary or partial view.
‘she caught a glimpse of the ocean’
The anonymous user noted that
_this may arguably imply even less intent than to "catch sight," since the latter can sometimes apply to an object one is specifically seeking.-
So perhaps the best option would be:
I caught a glimpse of my watch... shoot! It's 10:30!
(I'm embarrassed that I didn't put the two together myself!)