After much thought, I suspect the scratch came from playing with my dog who gets pretty hyper. She has really sharp claws. Unfortunately I didn't keep track of my purchase and I'm beyond the 60 days from date of purchase to be eligible for Apple Care. I got the watch on Jan 27th. Lesson learned. I also read a lot of forum discussions from various sites that Apple Care does NOT cover accidental damage like scratches. I've read people getting quotes from $199 to $269 to fix the screen.
Isn't there some protective anti oil/fingerprint coating on the glass? Maybe I just scratched that and the glass is OK. I may try the toothpaste trick if it doesn't damage the coating. If that doesn't work, I'll just slap a screen protector on it and hope it hides the scratch.
Apple does not cover cosmetic damage. And the coating on the Apple Watch display is the Olephobic coating. It acts as a protectant from minor abrasions and fingerprints. It does eventually wear off.
Also, toothpaste might remove some of the scratch off of the glass, but being your scratch looks fairly deep, I think it's likely it will still remain. And also in my opinion, no jeweler will be able to remove a scratch of that magnitude on the display of your watch, being it's glass and not Sapphire.
Glass has different consistencies than sapphire does when it's scratched and it's very difficult to remove, if at all, when the display is damaged such as yours.
I agree with the screen protector. Good luck with and let us know how it goes.