The Gemara establishes that the prohibition of carrying on Shabbos turns on whether an item is a burden (משוי) or an ornament (תכשיט): Shabbat 60:1 articulates that on Shabbos the Torah forbids carrying because of the concept of burden, so that an item without a functional seal is an ornament permitted in the public domain, while one with such a seal is a burden, and Shabbat 57a similarly presents the foundational distinction between items a woman may wear as ornaments versus items that constitute a forbidden burden.
The Gemara further refines when an item worn on one's person is considered nullified to the garment and thus permitted: Shabbat 58a rules that where an item is woven into the garment there is no rabbinic decree, while Mishnah Berurah 301:79 adds that even bells of gold are permitted when woven in, but anything loosely attached or tied raises concern that it may fall off and be carried.
Shulchan Arukh, Orach Chayim 301:7 codifies that one who goes out with an item that is neither a garment nor an ornament, carried in the normal manner of carrying, is liable, and that any ornament loosely fastened and liable to fall is forbidden; the Rama adds that a man may not go out with weapons or items that are not ornaments, and if he does he is liable to bring a chatat.
Shulchan Arukh HaRav, Orach Chayim 301:23 rules that attaching a functional utility item to one's garment does not permit it, since an item made for use in its own right and not serving the garment is not nullified to the garment even when the garment is far more valuable, and it therefore retains the status of a detached carried object.
Arukh HaShulchan, Orach Chaim 301:62 applies this framework explicitly to a watch, ruling that however beautiful or gold it may be, it is a utility vessel made to tell time, and since it would be stored if broken rather than worn, it is clearly not an ornament; therefore carrying it in a place without an eruv violates the prohibition by Torah law — a ruling whose logic applies equally to any wearable device whose primary purpose is functional rather than decorative.
Mishneh Torah, Sabbath 18:17–24 distinguishes between transferring a living person who wears clothes and rings — in which case everything is considered subsidiary to the person and there is no liability — versus a child with a purse hanging around his neck, where liability attaches because the purse is not subsidiary; this distinction between an accessory that is part of a person's normal dress and one that retains independent functional identity is directly relevant to a device worn on the wrist.
Beit Yosef, Orach Chayim 301:36 records that a proven amulet (קמיע מומחה) is permitted to be worn in the public domain on Shabbos because it is considered like an ornament for the ill person — comparable to one of his garments — and Shabbat 61b confirms from the Mishnah that one may go out with an expert amulet, with the Gemara clarifying that this applies even for a dangerously ill person; this analogy of a therapeutic or protective device worn on the body has direct bearing on a medical alert bracelet.
Minchas Yitzchak (Minchas Yitzchak 1:37) addresses the permissibility of wearing an electrical hearing aid in a public domain on Shabbos, comparing it to eyeglasses and walking canes and analyzing whether it constitutes a forbidden burden or whether the rabbinic concern that the user might remove and carry it applies, ultimately concluding that it is permitted if turned on before Shabbos and worn securely as part of one's clothing — a directly analogous ruling to a bracelet worn on the body.
Igros Moshe (Igros Moshe OC 4:81) addresses whether members of the Hatzalah emergency service may carry a small radio receiver on Shabbos to receive life-saving alerts, analyzing the Talmudic debate about whether carrying weapons on Shabbos is permitted as an ornament or forbidden as a disgrace — a framework with direct bearing on the status of a functional emergency device worn on one's person.
