Some developers created backported Play Store versions for Android 2.3–4.0, but these are unofficial, insecure, and likely non-functional today.
If already rooted:
Even if you manage to sign in, the Play Store’s backend will deliver an empty catalog. This is because the Google Play Store on Android 4.0.4 uses an old version of the Google Play Billing API (v2), while all modern apps require v3 or v4. The server essentially ignores your device. Android 4.0.4 Play Store
This paper examines the state of the Google Play Store (then transitioning from "Android Market") during the lifecycle of Android 4.0.4 (Ice Cream Sandwich). As the final iterative update to the pivotal Android 4.0 release, version 4.0.4 represented a period of stabilization for the Android ecosystem. This analysis explores the user interface design philosophy of the "Holo" era, the architectural changes in application distribution, the introduction of digital media content, and the security paradigm of the time. By understanding the Play Store of this era, one gains insight into the critical transitional period that moved Android from a nascent smartphone operating system to a mature, unified platform. Some developers created backported Play Store versions for
If you are trying to revive an old device, there are a few community-tested methods to get the store running: The server essentially ignores your device