I. Yes and No
Are you allowed to wear a regular (i.e. not gold) watch on Shabbos where there is no eruv (like in Brooklyn)? The standard answer in English manuals is no, but there is something on which the lenient can rely. For example:
Some permit an ordinary wristwatch with an ordinary band as well. However, a Ben Torah and one who is meticulous in observance of the Sabbath should refrain from doing so. (R. Gersion Appel, The Concise Code of Jewish Law, vol. 2 p. 302 n. 8)II. Jewelery and Clothing
Click here to read moreAccording to leading poskim, an ordinary wristwatch, even if not functioning, is technically a malbush that may be worn in r'shus harabim because it is attached to the wrist. However, it is generally agreed that this leniency should be avoided. (R. Dovid Ribiat, The 39 Melachos, vol. 4 p. 1352)
What could be the problem with wearing a watch? The only way it would be allowed is if the watch is considered either jewelery (takhshit) or clothing (malbush), otherwise you are carrying it. If it isn't a fancy watch and you wouldn't wear it if it wasn't working, then it can't be considered jewelery. That is generally the case with watches that aren't gold. And if you wear it under your sleeve, then it also might not be jewelery. That is the view of R. Nosson Geshtetner (Le-Horos Nassan 4:26), although most posekim disagree (cf. Piskei Teshuvos 303:5).
The question, then, is whether a watch can be considered clothing. On the one hand, who are you kidding? You wear it in order to know what time it is. However, R. Moshe Feinstein (Iggeros Moshe, Orach Chaim 1:111) writes that since you wear it directly on your body, it is considered clothing. This also seems to be the view of R. Shlomo Zalman Auerbach (quoted in Shemiras Shabbos Ke-Hilkhasah ch. 18 n. 111).
III. Why Not?
If that is the case, then why is it proper to be strict and not wear a watch? R. Moshe Feinstein writes that it is so that we shouldn't come to be lenient with other kinds of watches (pocket watches?) or watches that are worn on top of clothing. Note that he wrote that in 1956. I'm not sure how common such watches were then but today they are extremely rare, if they are worn at all. I'm not quite sure whether this reason to be strict still applies.
However, a different reason to be strict could be that there is room to disagree with R. Moshe Feinstein's and R. Shlomo Zalman Auerbach's view that watches which are worn on your body are considered clothing. R. Nosson Geshtetner (ibid.) strongly disputes this approach and rules that we are prohibited to wear a watch on Shabbos in a place where there is no eruv.
I suspect that there are posekim who reject R. Geshtetner's view and permit entirely wearing a watch on Shabbos. However, I haven't really looked hard or asked around. In practice, from what I can tell, people in Brooklyn generally only wear gold watches on Shabbos.
See also this post: link.