Even well before Byzantium and Christianity, “Greek Italy” was a fact of life, and of course back then nobody was making theological claims out of it. If Italy was Byzantinized at the time of Justinian (and I don’t know; as you might guess, this is outside of my areas of interest, as a non-Byzantine), he would not have been doing anything (culturally-speaking) that was not done long before him by the Greeks who settled in Italy in the year 600 B.C. with the founding of Neapolis (modern Naples). For an interesting overview of pre-Christian Greek settlement in Italy (excluding Sicily), which again I only bring up to contextualize the current conversation within wider patterns of cultural behavior common to Southern Italy since well before Christianity, see Michael C. Astour “Greek Civilization in Southern Italy” (Journal of Aesthetic Education, Vol. 19, No. 1, Spring 1985),
available as a PDF here: Please pay very close attention to the last paragraph, as that is the only one that does mention Christianity specifically, and the author makes the point that the vestiges of Byzantine Italy that remained in the extreme South in the early Middle Ages (by that time, a tiny portion of historically Greek Italy) were in fact a direct continuation or rather remnant of the previous Greek civilization that had been there.
So it would appear, if one takes the long view of history, that Wandile’s thesis that Southern Italy “became Byzantine” after Justinian only makes sense if it is meant in the same way that we could say that the Eastern Roman Empire itself “became Byzantine” by the founding of the new city and empire that had previously not been there (NB: exactly as Italy itself “became Roman”, which would be an odd thing to say). Otherwise, it is wrong. Southern Italy did not “become Byzantine” (Greek) – rather, it was always Greek (insofar as Christian history is concerned, since the Greek civilization was established there so long before Christ), and its portion and influence relative to that of the Romans/Latins shrank over the centuries, until the Greek Italians were left with one tiny corner. Shades of what happened to the Greeks in Egypt at Alexandria and elsewhere, I can’t help but notice… (the difference there being that the Greeks had established their city atop a preexisting Egyptian settlement, and of course that by any stretch of the imagination the Egyptian civilization itself is much older than the Greek, but I digress…)