The Ring app does not currently include an app for Apple Watch:
Ring - Always Home on the App Store
If you would prefer that it did, then you may wish to submit feedback to the developer.
Notifications may still be pushed to your Apple Watch, depending on your Notifications settings and on the settings and status of your devices.
With Notifications for the app enabled and with Wrist Detection also enabled, in order to bring notifications to your attention as soon as possible:
- If your iPhone is unlocked, then you will receive notification alerts on your iPhone instead of on your Apple Watch.
- If your iPhone is locked or asleep and your Apple Watch is unlocked (regardless of whether the watch display is asleep or awake) and on your wrist, you will receive notification alerts on your watch.
- If your Apple Watch is locked or has Do Not Disturb enabled, then notifications will go to your iPhone.
Notifications on your Apple Watch - Apple Support
See and respond to notifications