In the right circumstances, of course it could. I don't see the right circumstances likely to arise in Late Antiquity or after it. If anything, the Hellenistic and Roman times were the point when the foundations of this shift would have been laid. For various reasons (not all entirely clear, though clearly the pre-Hellenistic prestige of Aramaic played a role), it did not happen to a sufficient extent before the local languages arose as cultivated varities tied to local identity (and to specific religions, very often).
The fact that Greek disappeared relatively quickly from most uses in most of the Levant and Egypt after the Arab conquests (not without leaving an imprint) while Aramaic emphatically did not (to a lesser extent also the case for Coptic) suggests a relatively shallow rooting.
Remarkably, by the ninth century many Arabic thinkers of all faiths regarded Greek (they called it "Ionian", as in, Classical Greek) as an utterly dead language, even the very few who were proficient in it. They did no see continuity with Byzantine Greek, which they usually called "Roman".
The point is prestige. Between Alexander and the time of the Severans, Greek was unquestionably the prestige language in most of the Levant, main exceptions being Jewish milieus (important but somewhat self-contained) and some peripheral areas (Palmyra, the Nabatean realm, Osrhoene and Hatra both often under Parthian suzerainity, generally the arid hinterlands). However, it seemingly failed to take root as a spoken language outside urban areas and probably even there mostly among the elites, as opposed to what had begun to happen in many parts of the West for Latin and Greek itself in Anatolia. Then, again unlike the case of the West and Anatolia, Christianity and Late Antique religiosity more generally came and Aramaic(s), Coptic, Armenian and, to a small degree, Old Arabic (as well as Gothic and Ethiopic) became, or became again, prestigious written languages, tied to faith and local identity, carrying prestige. Not necessarily in opposition to a Roman overarching identification, but certainly with some level of contrast with Greek. Then, Greek had lost its main chance to take root.
Compare with Gaul, where, while Gaulish being widely spoken into Late Antiquity and even occasionaly written, it lacked prestige and nobody seems to have considered, for instance, translating Christian scripture into it.