Because Google has discontinued Play Services support for Android 4.0, even if you successfully install the APK:
Since the built-in browser on Android 4.0 is outdated, you may need to download the file on a PC and transfer it via USB, or use a modern browser if one is installed. play store apk android 4.0
: Added more granular settings for when and how apps should update. Current Limitations Because Google has discontinued Play Services support for
: Grouped content (Apps, Movies, Books) into distinct categories with larger images. Wishlist Support : Introduced an easy way to save apps for later. Auto-Update Improvements play store apk android 4.0