May I wear my watch on Shabbos?
When in a public domain where there is no eiruv, we may only ‘carry’ the clothing and jewelry we are wearing (though the halachic parameters of wearing jewelry are complicated). Everything else is considered a masa (load), and may not be carried due to the melachah of hotza’ah (forbidden act of carrying) on Shabbos.
Watches don’t assume the halachic status of jewelry according to most opinions and may not be worn in an area without an eiruv. There’s a minority view that considers a wristwatch a garment, but many poskim reject that position and maintain that it is a keili (device) for telling time. The exception is—according to many opinions—a woman’s watch that is made of gold or is otherwise exceptionally elegant, that would be worn even if the watch stops working, provided it is not worn under the sleeve.
There are people who are particular not to wear a watch on Shabbos even indoors or in places where there is an eiruv, either because they consider it at odds with the kedushah (sanctity) of Shabbos or due to the concern that they may inadvertently wear it in a forbidden area as well. This is similar to the halachah that we should avoid putting things in our pockets on Shabbos as a precaution against carrying.
https://asktherav.com/article-574-may-i-wear-my-watch-on-shabbos/ See also: https://asktherav.com/2940-are-all-watches-permissible-to-be-worn-on-shabbos-are-they-muktzeh/