Just to be clear, there was never a Greek empire. There were, in the course of the Classical period, a Spartan hegemony, an Athenian empire, a Spartan empire, a Syracusan hegemony, a second Athenian empire, and a Theban hegemony. These power blocs were each eventually broken up by their rivals or by outside forces. All of mainland Greece besides Sparta was eventually subjected to Macedon in 338/7 BC.
The history of the mainland Greek states after this subjection was one of them coming to terms with a new world order dominated by powers much greater than theirs. The kingdoms that emerged in the wake of Alexander's campaigns and the Successor Wars encompassed vastly greater territories, and mustered vastly greater financial and manpower resources, than any Greek city-state could match. While the Greeks mostly tried to deal with this by establishing alliances and coalitions, they never banded together as a single empire in order to carve out their place in the new Mediterranean anarchy (as Eckstein famously characterised it).
By the time the Romans seriously turned their attention to Greece, the major powers in the region were the rump kingdoms of Macedon and Epiros, as well as Pergamon, Rhodos, the Aitolian League and the Achaian League. After the first phase of Roman intervention (c. 202-188 BC), in which Macedon was decisively beaten, the Greeks were left nominally independent, but clearly subject to the whims of Rome. By this point Rome already dominated all of Italy and the entire Western Mediterranean; there was no chance that even a coalition of all the Greek states could have mounted an effective resistance.
Even so, the Greeks continued to wage their own aggressive foreign policy from their several states, and frequently proved a nuisance to the new superpower of the Mediterranean. It took a series of destructive campaigns for Greece to be formally subjected to Rome, a process that was completed only in 86 BC when Sulla besieged and sacked Athens. Under the Augustan establishment of the Empire, mainland Greece became the province of Achaia - devolving into a peaceful backwater over the centuries of the Pax Romana, far from the war-torn borders of the empire.