You are posting to the forum of Garmin Connect Web, but it looks like you are referring to the Garmin Connect Mobile app for iOS, or am I wrong? In iOS, if the notification counter cannot be cleared from the GCM icon on the home screen, then you can try forcing the app to stop, and restarting.
It does not seem to be related to the GCM app. I just re-tested it on my Instinct 2, and the notification clearing works both ways - when I clear a notification on the watch, it instantly disappears from the phone too, and vice versa as well. Tried with app notifications, as well as with calls. So I'd tell it is either a problem of some missing permission on your iPhone, or a device related problem - you may be better advised on the subforum dedicated to the model(s) you own.
Yep. Mobile app. RING notifications will not clear on iPhone
Moved thread to correct forum