In my opinion, "doesn't" must be a better choice though your watch may out of work for a long time or for a while. Because you are mentioning an action in the present, hence, there's no need for you to use " won't". Moreover, "doesn't" sound more smooth
"Won't" is a contraction of "will not." "Will not", as written here, is in the form of a prediction. In addition, by using "won't", you are giving free will to a watch.
Indeed, and in the utterance we are talking about, 'won't' is unlikely. However, If I have been having problems with my computer, and see it as a malevolent beast forever trying to frustrate me, I am very likely to say: