;-)
Alexander’s conquests were centred further east than the Roman Empire was. Alexander’s conquests which later fell to Rome included Macedonia (of course), Greece, Asia Minor, Armenia (although that was always disputed), Syria, Phoenicia, Judaea, Philistia, Egypt and Cyrenacia. These parts all used Greek as their lingua franca up to the time of the Muslim conquest, after which Greek was gradually driven back to its cultural hearth in Greece.
Everything west of Alexander’s domains used Latin almost exclusively, although Greek and Phoenician traders from the east were typically accomodated in Greek. The impact of those traders on the population at large was basically nil, though, and as trans-Mediterranean trade was strangled by the later emperors, Greek was essentially forgotten in the Western Empire, well underway as early as the time of St. Augustine, whose teacher, St. Ambrose was regarded as a man of profound learning for his fluent knowledge of Greek. St. Augustine’s north Africa west of Cyrenacia, used Latin as well, until the Arabs burned the successor Vandal kingdoms essentially to the ground.