Stolen watch

Not sure if this will help with your specific question, but see Garmin's Support site:

• I’ve Lost My Garmin Device | Garmin Customer Support
• My Garmin Device Was Stolen | Garmin Customer Support

Thank you for your reply. I saw the articles but this is not what I want. 

I want to block the watch in a similar way you block a credit card. 

They can't send a kill signal to the device. Perhaps they could prevent it from being used on another account, but I doubt it. I suggest contacting Garmin Support directly via phone call.

I don't want a kill switch, as this will only work if the watch still has internet connectivity. 

Garmin should not allow a device to connect to Garmin services if a specific s/n was declared stolen.

Which is why you'll need to contact Garmin directly. Preferably by phone or chat for a more immediate response. Good luck.  

FWIW, there was a similar conversation a few years back - Hello. I am the owner of a Garmin Fenix 5 x plus This watch was stolen from me

If you contact Garmin you should have some prove it's stolen, like a police report.