I deleted an app but when I try to reinstall it on the App Store, it still says open

"Open" in the App Store indicates that the app is already on your device.

Have you tried using Search on the device? See: Use Search on your iPhone, iPad, or iPod touch - Apple Support

Also note that apps can be removed from the home screen without being deleted. Go to the App Library (swipe left past your last home screen). From there you can touch and hold an app and slide it left back to the home screen.

Also note the Search window at the top of the App Library.

You cannot redownload it because it was not deleted. It is still on your device.