Apple Watch cover falling off

If the watch needs repair Apple will usually provide a service exchange device and sends the defective watch to a central reconditioning facility for full overhaul. The watch is more complex than it looks, and even simple repairs are not practical in-store.

It sounds like like you need to Contact Support (link) for more advice and to setup a support case for your broken watch.

I think they send it on to an authorized repair shop. You could call and ask an Apple employee.