That's exactly the case. When Alexander the Great conquered the Persian Empire, that meant an empire that spanned from the Balkans, to Judea, to Egypt, all the way to India--and after the death of Alexander his empire was divided among his generals, and under their rule those conquered areas became Hellenized--Greek culture, Greek language, Greek religion. Some areas revolted from Hellenistic rule, such as the Jews under the Maccabees, but even still, Greek had saturated many levels of Jewish life, especially among Jews who made communities elsewhere, such as in Alexandria, Egypt, where a group of Jewish scholars translated Jewish holy books into Greek, a work we call the Septuagint or LXX (both mean Seventy, because traditionally it's believed 72 Jewish scholars worked on the project).
Those Greek kingdoms also established colonies in parts of the Mediterranean, such as in Sicily and southern Italy. Greek had already been a powerful cultural and language force for centuries, Rome itself in a sense had been influenced by the Greeks for centuries before Julius Caesar, such as can be seen in the Latin alphabet, which is based upon an archaic, western Greco-Italic script (the Romans weren't the only ones who did this, the earlier Etruscans also borrowed the Greek script to write their language).
So when Rome, under Julius Caesar and Augustus, effectively unified the entire Mediteranean under Roman hegemony, there was already a very well established language that was known by most peoples, that was useful for trade, for mutually understandable correspondence between disparate groups--Greek. Specifically the vulgar or common Greek spoken by craftsmen, traders, merchants in the marketplaces, the international language of the common people; or Koine, literally "Common".
Which is why when early Christians established communities from as distant regions as Spain and Syria, there existed a common language, and they could write to one another. And so when St. Paul wants to write to the many different communities of Christians, he has those letters written in Greek, when Paul preached, he preached in Greek. When the story of Jesus was written down, Christians wrote it down in Greek, as the four Gospels of Matthew, Mark, Luke, and John.
It's the only language it would make sense to have wrote in. No matter what one's first language was, Greek was the language almost everyone knew.
So it's exactly the case that Greek was the common tongue of the time, and that it's the language of the New Testament. Indeed, to have written in any other language would have been absurd--especially in a functionally near-dead language which Hebrew was at the time.
-CryptoLutheran
