Q. Is one permitted to wear an electronic watch with a push button LED light on Shabbos?

A. Biur Halocho (308: 3) deals whether we define a keli shemelachto leissur by the majority of its use. In the case of an electronic watch with a light feature, since most of its use does not require that light, it would be permitted (Halocho Arucho p. 21). Halichos Olam (4, p. 265), also Yichave Daas (2: 49) opine that we cannot legislate and create new prohibitions against wearing such watches out of concern that one may press the buttons. Thus, it is permissible to wear a battery-operated watch, though it is forbidden to press the buttons or to adjust the time. See also Rivavos Efraim (7: 104: p. 81) who in principle is lenient in regards to the watch becoming a bossis or a base to an item (battery etc.) of prohibited use. See similar in Beis Avi (3: 91) and other Poskim in Hachashmal Behalacha (2: p. 325- 328) that maintain a lenient view.
Menuchas Ahavo (24: 36) quoting Tefila Lemoshe (1: 23) in principle also permits it’s use, mentioning that some place tape covering the buttons, to avoid using them on Shabbos. However, he also quotes Poskim who are stringent, since the electronic watches keep on developing all kind of computerized features and on-line functions, that are enticing to use (See question on Apple watch 3).
Shalmei Yehuda (p. 49) quotes Horav Eliayshiv zt”l opinion, that the watch is muktze since we are concerned he will press buttons. However, Horav S.Z.Auerbach zt’l was lenient, when one is sure he will remember not to turn on the light or other features.
Horav Shlomo Miller’s Shlit’a opinion is that the use of the watch is permitted, but it is preferable to cover the buttons.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a