The short answer is no, automatic watches do not have batteries.
An automatic watch contains a purely mechanical movement that is self -winding, thus it is automatic.
For a complete explanation of an automatic watch movement and how it operates please see my blog post here.
The type of watch that contains batteries is a quartz watch. Quartz movements are completely electronic, and battery powered. I have written a post about the differences between solar quartz movements and automatic movements, this will give you a great idea of the differences and similarities between quartz and automatic movements.