Apple Watch stopped charging

When you put the watch on the charger, make sure you're seeing the lightning bolt indicating a charge if completely dead, or if there's still a bit of charge the whole screen should be a partial circle with the lightning bolt.

If you don't, pull the watch off and try reseating it on the charger and maybe rotating the it to a slightly different angle. Sometimes I have to put mine on the charger a time or two before it "takes"

I would clean the underside of the watch and the curved part of the charger. I've used rubbing alcohol in the past to get off anything that has built up on the watch from touching skin.

Since you've already tried different port with the same results, I'd say it may be time for a new charger cable. Bring your old cord with you when you go to the Apple store.