If the watch doubles as a computer, telephone, flash light, etc., why wouldn’t it be considered mukzeh? Has anyone ever poskened that one can walk around with such devices? Can you walk around with a laptop computer? or a flashlight? or a mobile phone? Unless you hold that work done by electrical systems is not prohibited on Shabbos (something often held by non-frum Jews who accept the idea of Shabbos, but not any inconvenient details), this should be an obvious “no”.
A regular watch (no electronic settings, just a battery moving a mechanical gear – or an all mechanical one with a spring moving a mechanical gear) is a totally different machine.