The Omega Speed Master, which is arguably the most famous watch worn in space, runs on an automatic movement. It has worked and still works great even in space where there is very little gravity.
However, the astronauts usually manually wind the watch more often than they would on earth.
This goes to show that an automatic watch will work in space.
Of course, on Earth, the downward vector of gravity does help to keep the watch wound.
In space, however, there is no friction from atmosphere, so the moving parts in the watch move more freely.
Do automatic watches work in space? Yes, they do. Probably right now!