In most cases: no.
Even if the Play Store opens, it will fail to download apps due to outdated SSL/TLS security (required by Google’s servers).
What you might see:
Android 4.0.4 was released in 2012. Google no longer supports the Play Store on this version (support ended around 2018–2019).
If you are trying to use a 4.0.4 device as a daily driver – stop. This guide is for hobbyists, retro device fans, or offline use cases.
If the official Play Store is dead, how do you get apps? The answer lies in third-party markets and archival repositories.
By modern standards, the security model of the Play Store in the Android 4.0.4 era appears incredibly permissive.