Can an automatic wrist watch run in space?

By 'automatic', do you mean self-winding? And do you mean in a state of weightlessness? I believe it would, as long as someone is wearing it. Wrist movements cause a self-winding mechanism to work based on the inertia of the weights within the watch. Inertia is a property of mass, and it does not go away in the absence of gravity. Battery powered and ordinary manually wound watches would also work, I believe.