I wouldn't go that far, but it is very unlikely. Arabs are the most likely early governors, so why would they need to learn to speak and write a new language?
In the Rashidun era, few tribes had left Arabia, and many Arabs were Christian or Jewish converts or their immediate descendants. Local administrators were kept for the most part. The conquering soldiers would have spoken Arabic, but as many of the conquered people didn't, it would have been helpful and feasible for the new rulers to use the old languages for secular purposes. It probably actually happened, just not across the entire empire. In this early, malleable stage a Caliph who adopted Aramaic for secular purposes, keeping Arabic for religious purposes and in Arabia itself, probably could have seen that decision stick.
In the Umayyad era, more Arabs were settling in, but in the early era they still used local administrators. The same thing could have happened here under an open-minded Caliph. Plus the capital was in Syria, the heart of Aramaic-speaking land. That being said, in the late Umayyad era, where non-Arabs were made second-class citizens, this scenario would be impossible.
If the Umayyads take Constantinople, that's a lot of casualties and adopting regional languages when there are not enough Arabs becomes a necessity.
In the Abbasid era, more people were being tolerated and the Abbasids were providing more autonomy and respect to non-Arabs especially if they converted. Perhaps, in attempt to get a lot of people in Mesopotamia and Syria to convert at once, they preach in Aramaic and give some power back to deposed bureaucrats during the Umayyad era. This solidifies Aramaic as an important language in Syria and Mesopotamia, and if the Abbasids lose the rest of their empire as they did in reality, Aramaic could become the most important secular language. Arabic remaining for written scripture.