How did the Persian Wars unify Greece?

Answer 1

To defeat the Persian takeover, the Greek city states stopped fighting each other and banded together to fight the common enemy. When the Persians withdrew, this temporary expedient unification dissolved, and the city-states went back to fighting each other.

Answer 2

The southern Greek city-states unified briefly to repel the Persian ivasion in 480-479 BCE, then went back to their usual practice of fighting each other.

Athens then organised a coalition of city-states to continue opposing Persia, but progressively turned it into serving its own interests, effectively turning it into an empire of its own. The Dorian Greek cities then formed the Peloponnesian Leage to counter this.

Unify? No.