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May one wear a smartwatch on Shabbos and Yom Tov?
Obviously, when touching the screen or buttons, as with all electronics, this falls under multiple forbidden categories of work (Melachot). These are Boneh - Constructing, Makeh Bepatish - Completing an Item, Mav'ir - Kindling a Fire, and Molid - Creating a new entity (as explained in Chazon Ish, Orach Chaim 50:9).
However, even when not interacting actively with the smartwatch there are multiple issues.
Foremost, if the watch wakes when you raise or turn your wrist (as many Apple watches do), there is an issue called "Pesik Raisha Denicha Leh - פסיק רישא דניחא ליה" which effectively means doing an action that inevitably causes a Melacha you desire to occur. This is forbidden on Shabbos and Yom Tov (Gemara Shabbos 75a, Shulchan Aruch 337:1).
Next, even if the screen is set to 'Always On', almost all smartwatches on the market today are designed to monitor and measure various body functions, such as one’s blood pressure, step count, or heartbeat rate, etc. while others are designed to monitor and measure weather information, etc. These Melachot, whilst being indirect, are also both inevitable and beneficial and as such forbidden. In addition, there is the added issue of the digital display often being updated based on the wearer’s hand movements and vital information. Therefore, unless it is a medical necessity, these should not be worn.
In all cases when the watch may not be worn, since the watch has no use on Shabbos and Yom Tov, it gets a status of Muktzeh and may not be moved at all (see Shulchan Aruch 308:7 the laws of Mukteh Machmas Gufo).
If, mistakenly, one finds himself wearing such a watch while in the street, they do not have to remove it immediately, but may go home and remove it at that time (Rav Y.S. Elyashiv, quoted in Me’or Hashabbos, vol. 4, pg. 337).
If, however, a smartwatch displays the time at all times, doesn’t require activating, and all of the other sensors and functions (e.g. GPS, activity, heart rate, etc.) are deactivated, one may wear it on Shabbos and Yom Tov. In the same vein, non-smart digital watches (e.g. older Casios with no sensors at all) may be worn, provided that they do not require a button press to display the time.
Even when permitted, one should only wear a watch in a place where there is an Eruv. This is because whilst some treat a regularly worn watch as clothing (R. Shlomo Zalman Auerbach as quoted in Shemiras Shabbos Kehilchasah ch. 18 n. 111), most opinions rule strictly and forbid wearing non-Jewelry watches (i.e. a watch you would not wear if broken) in places without an Eruv (Mishna Berura 301:45, Ketzos Hashulchan 115:28 and many others, see Iggros Moshe Vol 1:111).