Apple Watch app missing from home screen on iPhone

If you can open it from the App Store, then it IS on your iPhone. Just not where you want it.

It is either on another screen (swipe left), or it is in a folder.

You can use Spotlight to search for the "Watch" app

Use Spotlight Search on your iPhone, iPad, or iPod touch - Apple Support

This is a trick you can use to put an alias for "Watch" on your Home Screen. Once Spotlight search has found the "Watch" app, you press and hold on the "Watch" app icon in the Spotlight search (a popup menu will appear, IGNORE IT, and keep holding the icon), now drag the "Watch" icon and you will be able to place the icon on your Home Screen. This is not a 2nd copy, but an alias that points to the Watch app.

You can use this trick to have lots of aliases for apps that you find you want to have on multiple screens, or in different folders. For example, you can have a collection of common apps in a folder, but there is one you use all the time and ALSO want an icon for that app on your Home Screen.

Move apps and widgets on the Home Screen on iPhone - Apple Support

Try and swipe all the way right to the App Library. Search for the watch app there, hold it. And press add to Home Screen to get the watch app back on your homescreen