This on an Apple Watch 6 using just-released OS 8. My companion iPhone is a model 12 using iOS 15.
Searching the web results in instructions on how to do this in previous OS versions (Settings>General>Wake Screen>auto launch audio apps), but there is no Wake Screen setting under General settings in watch OS8.
Annoying as I must press the Digital Crown to dismiss the app (not quit the app) to return to my default watch face. Also the audio app indicator-a white dot a top of watch face) blocks the red dot notification indicator.
Watch app (on the iPhone), tap “Display and Brightness”. Turn off “auto-launch audio apps”. Or in the same place in the Settings app on the watch itself.
Many thanks to D. Miller for the prompt, accurate solution. FYI while this prevents the auto launch of Now Playing app, it does not prevent the appearance of the white dot at the top of watch facing indicating music is playing on a device on the same network as the watch—in my case HomePods—and blocking the red dot Notification indicator. However if a notification received on watch, the red dot notification will briefly flash upon wrist raise then fade, replaced by the white dot.
If you don’t need it, what might work is to delete the “Now Playing” app from the watch. Press the crown to show the app screen (I have changed mine to be an alphabetical list rather then what I consider the inscrutable screen of tiny round icons with watch app / app view, but you can do this either way), long press on the Now Playing icon or listing, choose edit apps, and tap the small delete control on the icon or listing (the icon for now playing, which you may recognize from the white dot, is a white circle with four red lines forming a triangle pointing to the right.)
Implemented your suggestion which does eliminate the auto-launch of Now Playing app, but the playback control white dot at top of watch face persists, even after restarting both watch and iPhone. Assuming this function is baked into either the Music player app or iOS 8 itself. A First World problem that is annoying but tolerable. Many thanks again for your suggestions.