First we must remember that Alexander did not live very long, it takes a long time for Empires to merge and change cultures, which all the Hellenistic Kingdoms did to very different degrees. But regarding Alexander himself:

Embracing foreign cultures: He married multiple times in the east to non Greeks, most famously Roxana of Bactria but he also took Persian brides.

Mass Wedding: Arrian records that Alexander held a mass wedding for his troops in Susa (although to varying success), illegedly 10,000 strong.

Although I cannot find the quote from Arrian, he also started to dress and act more Persian, much to the horror of his men. I believe trousers were worn (true Greek men wear skirts).

So as you can see Alexander was quite the fan of Eastern culture and certainly did not try and oust beliefs. Was Greek culture imposed by him? Not especially, although he did build multiple cities (many named after him, one way or another). The successor states also tended to embrace local culture a lot, notably the Ptolemies were Pharaohs and practiced Incest like their predecessors. The Seleucid Kings started to dress like Persian kings, and many had beards (which Greek men as a rule didn't, it was considered barbaric).

Greek culture (notably theatre!) and religion did permeate the east, but it is easier to assimilate a nobility into the local culture than a conquered people into a foreign one.