Philipos II of Macedon lives longer, carries out his expedition against Persia, and accepts the deal that his son declined in OTL. Basically, Anatolia, Syria, the Levant and Egypt are ceded to him in perpetuity. He starts consolidating. Meanwhile, the Akhaimenids are humiliated by this shameful defeat, and Persia collapses into a renewed bout of civil war / succession struggle. (And they just
had one.) When Philippos dies, the Greek poleis briefly rebel, but Alexander gives them a beating and fully incorporates them into the empire his father left him. Instead of going East, he then heads West, annexing Megale Hellas into the Empire. Carthage offers him a nice sum in gold and a lot of flattery, and as a result, he leaves them alone. They have to accept his claim to all Greek colonies in the Western Med, though. (Most notably, Massalia.) Alexander then spends some time warring in Italy and Illyria, eventually annexing all of those areas into his Empire. He also annexes the Kingdom of the Cimmerian Bosporos, and endeavours to turn the Black Sea into a 'Greek Lake'.
Congratulations, the rise of Rome has been averted, and this whole region is now governed by a single Greek empire:
All the newly-added regions had already seen quite some Hellenisation, but this is now ramped up. Regions that are densely populated and distinctly un-Greek have been deliberately left out, so as to avoid cultural syncretism that would run counter to the premise of this thread. We want this Empire to be unquestionably
Greek. Well, give it a few centuries, and what I've outlined here will be completely Hellenised.