Can someone wear a self winding automatic watch on shabbos as long as its ticking? Are there any stringent opionions?
There is no problem in wearing a self-winding watch while it is ticking. The watch is already ticking, and wearing it does not cause anything to happen. Even concerning actually winding a watch some are lenient, and although the custom is to be stringent (based on the ruling of the Mishnah Berurah), this appears to be only out of concern for the honor of Shabbos, or lest one winds the watch when it has stopped — concerns that don’t apply to a self-winding watch. See https://dinonline.org/2011/07/13/winding-watch-on-shabbos/.
Sources: See Shemiras Shabbos Kehilchasah 28:28 and sources cited in the footnote. I do not know of stringent opinions in this matter.