I have a digital watch. Can i reprogram it?
Not with an Arduino and probably not with any means at your disposal.
What do you want to do with it ?
Depends, is it like this one:
In answer to the original question the answer is Yes, as long as you carefully choose which digital watch you buy to re-program.
I've written a brief article on my blog about the variety of fitness tracker / smart watches which are available which can be reprogrammed (if they have the nRF51822 processor)
http://www.rogerclark.net/nrf51822-based-fitness-trackers/
and this youtube video I made a few days ago - YouTube
(Moderators, sorry is this is a cross post, so perhaps delete the post to the older thread about the same subject if you'd rather this information is only in one post - and keep this one as its a far more recent thread)