Alexander had a dream of cross-cultural change, but none of his successors did or at least none gave the idea much practical application. Hellenic culture didn't rub off to the Egyptians, Persians or Syrians to a significant degree (though there were some influences, wich should not be underestimated)

We speak of a Hellenic age because Alexander made lots of changes possible: adoption of a standard hellenic tongue (Koine Hellenic, based mostly on the Attic dialect), Greek dominance over the entire eastern mediteranean, flourishing trade and increased contact with (but not willing adoption of) eastern customs, scale enlargment in virtually every economic area, major geopolitical changes etc. Wether we take Alexander himself or his death as a line between previous and after is a minor point, and not really accurate anyhow.