Short answer, sort of. Not nearly on the same scale, as suddenly, or directly. However, it can definitely be counted into the contributing factors.

Alright let's start with the basics. There was no "Greek Empire." There was the Delian League, which was basically the cover for Athenian imperialism during the Peloponnesian War in the 5th century BCE. That ended when Athens was defeated by Sparta and the Peloponnesian League in 404 BCE. In the 4th century Alexander the Great briefly ruled a Hellenistic Empire before it collapsed into a collection of smaller empires and kingdoms after his death. Then there was the Byzantine Empire which could be described as a Greek empire, but they would have called themselves Romans before being slowly conquered by the Ottoman Turks in the 1400s (finally ending with the fall of Constantinople in 1453). You couldn't really attribute any the fall of any of those to disease; they were all defeated or broken up by military means.

There is an argument to be made that Athens could have established an Athenian Empire via their Delian League if they had won the Peloponnesian War, which seemed possible before the Plague of Athens in 430 BCE. As it was 2400 years ago we don't know exactly what disease it was, but it crippled Athens and led to a brief period of anarchy within the city that provided Sparta with an opportunity to turn the tide of the war against Athens. However, it did not have the same effect as the Old World pathogens on Native Americans (where I've seen estimates as high as 90% of the population succumbing to disease). Athens was not depopulated to the point of not being able to withstand Sparta and the war continued for 26 years and the effected populations would more or less recover.

Shifting to the Roman Empire, or more specifically the Western Roman Empire (as the eastern portion became the Byzantine Empire described above). Famously the most direct factors of Roman decline were external invasions and the breakdown of central imperial authority allowing the Germanic tribes in the empire to carve out their own kingdoms including the Ostrogothic conquest of Rome itself. Disease was a major contributing factor to getting the Romans to that point of weakness though.

In the mid 2nd century the Roman Empire was hit by the Antonine Plague. It was probably an outbreak of small pox or measels that was brought back by soldiers on the Empires eastern frontiers. The plague had a casualty rate as high as 25% and according to Cassius Dio caused 2000 deaths each day in Rome at its peak. Because the plague started in the Legions it was particularly rampant in their ranks allowing the Parthians to launch a successful invasion of Armenia in the east and a number of Germanic tribes to make incursions over the Danube River in the north. The weakened state on the frontiers was one of the factors that contributed to the chaotic span of invasions, civil wars, separatist kingdoms, and political struggles that are collectively referred to as the Crisis of the Third Century. A major part of the Crisis was actually another plague called the Plague of Cyprian from 249-262 CE. The plague caused manpower shortages across the Empire and contributed to the environment that allowed so many attempts at usurping the imperial throne in such a short span. The Crisis ended with the rise of the emperor Domitian and the empire can even be said to have recovered slightly under his successors in the 4th century, but Rome was never really an expanding conqueror again and some of the innovations of the Dominate system of government and reorganization of the military by Domitian can be seen as contributing factors in Rome's eventual fall, but again not as directly as in the Americas.

So they aren't really comparable. It's not even all that likely that these were the first time the Greeks and Romans were exposed to these diseases like the Native population of the Americas in the colonial period. Roman decline was certainly influenced by disease, but it was not on nearly the same scale as Old World pathogens coming to the Americas for the first time.