What would have happened to the West?

Greek culture wouldn't be as relevant and we might well have lost some philosophical treasures (though actually the best of Greek philosophy pre-dates Socrates and is badly known, and also Greek influence in Italy was already very strong) but Rome would have existed and conquered Italy with even less problems than they had with the Hellenistic intervention of Pyrrhus. Still, Carthaginians were kind of allies of Persia and that could have conditioned the Roman expansion beyond Italy. It's difficult to say because Persians weren't good sailors nor had any excellent army either. Also, as the empire of Alexander, that of Persia might have fragmented.

My guess: much harder Punic Wars, with better chances for the Carthaginians and the question of how well Persia would have managed to dominate the Eastern Mediterranean against whichever of the western powers would have won the Punic wars.

Maybe we would be all now Mazdeists or Mithraists... who knows?

Although Persia and Carthage had their alliance, it was more borne out of the difficulties they both faced in dealing with the Greeks at opposite ends of the Mediterannean. With the Greeks out of the way, the two civilization would be in direct competition for trade, territory and tributary states.

My theory is that Rome, already having received alot of Greek culture and beginning to assert itself in Italy, would have played the two masterfully to Rome's own benefit. Both sides would have attempted to secure Rome as the valuable ally she undoubtedly was. I personally doubt the Persian system of satrapy was advanced enough to extend much further than the southern Balkans, the Scythians had successfully proved this on land and the Persians were largely reliant on subject states for sea power. In the end Rome may well have had an easier time of subjugating Gaul, Spain and the Balkans while Carthage and Persia slug it out in the Central Mediterannean. Playing the two off against one another until the moment was right, Rome would then swing south and take out one of the two before moving on to the other after finding some pretext. Both Carthage and Persia were vulnerable to Roman intervention, two empires which comprised large subject populations always ripe to rebel.

What if Alexander wouldnt have died and continued his quest to the west?

Originally posted by avesta If the greek resistance during the Achaemenid period had been suppressed, what would have become of the Western world?  Would it have become a vassal of eastern kingdoms under Persian rule?

All depends on the Romans and their ability to built a world empire. The Persians had no habit of changing the culture of their subjects, although it was exactly under their reign that Mesopotamia began to lose its identity as a distinct civilisation. But Greek culture was clearly on the rise and the best the Persians could have hoped for was political control, but not cultural impact. That means it would have made hardly a difference.