That would be a remarkable achievement if nothing else. However, let’s analyze it a bit.
There were 3 great powers in the region after the Persian wars. Athens, Sparta and the Achaemenid Empire of Persia. Just because the Greeks managed to defeat the Persian expedition, doesn’t mean they were defeated or subjugated.
Nevertheless, both Greek city states had great commanders and politicians, leaders that worked great together. Keep in mind that after the defeat of the Persians in the battle of Plataea, the Greek alliance counterattacked liberating Ionia (Asia minor back then) since the cities there were actually Greek, primarily Athenian with a bit of Spartan in the southern areas, Cyprus (same case as before), and Byzantium.
Back in 481 after Xerxes first sent ambassadors to the main city states apart from Athens and Sparta because they were big ones and he knew that in order to defeat them, the smaller cities would have to fall first, a confederate alliance of Greek city-states was formed which had the power to send envoys to ask for assistance and to dispatch troops from the member states to defensive points after joint consultation.
Only 70 of the nearly 700 Greek city-states sent representatives. Nevertheless, this was remarkable for the disjointed Greek world, especially since many of the city-states present were still technically at war with one another.
Keep in mind that we are talking about a pre-diplomatically advanced world, completely unlike our modern day one, where simple things like miscommunication could provoke war. Although it wasn’t an all out one, more like skirmishes and a few actual battles.
That single point in history was the first true alliance and first step towards a unified nation. Herodotus called the union “the Hellenes” (the Greeks), and the Greeks that swore alliance.
Now in this alliance most of the important city states of the time were part of, however, while their primary goals were the same, their secondary ones differed.
Following the Greek counterattack and liberating all Greek city states that were previously under Persian control, Sparta wanted to stop advancing and withdraw. They were always the cautious ones. The ones who thought things with a military purpose first. They knew that advancing further inland was a suicide and that defending all the coastal cities in Asia Minor was difficult to say the least. That is why they proposed transplanting all Greeks from Asia Minor to mainland Greece.
Of course, Athens denied their proposal due to its nature. Athens always favored expansion without regards to safety and security, they didn’t often think before taking the leap and that proved costly later on.
That was the first spark that lighted the fire of disagreement between these two great cities. After Sparta backing out, Athens became the sole leader and it got into its leader’s head that they were the best.
Had they actually listened to the Spartans, the union would last way longer and it is possible that unless they accepted Alexander’s proposals or he managed to enforce his opinion upon them, then we would have no empire to the east and way more battles between Persians and Greeks in the future. Perhaps it is possible that had Alexander stayed in Europe, then the Romans would have failed since they would face a united Greek alliance and not single cities inviting them in and offering their enemies while providing the necessary resources at the same time.