No, that is a profound misunderstanding of the situation in Southern Italy at the time of unification. There was an important Greek cultural influence in Southern Italy, but it stopped being tangible by the twelfth century. I wrote a very cursory answer on Southern Italy and the Italian unification process here. You might be interested in the following part, where I talk about the situation in the eleventh century, after a group of ethnically Normal soldiers of fortune usurped the (Lombard) power structures:

Indeed, Greek Christians (who had in great part welcomed the Normans) played a key role in mediating between the upper class of Latin lords and the subject populations of Muslim serfs (according to Johns J., The Monreale Survey: Indigenes and invaders in medieval west Sicily). It's interesting to note how within a generation of the Norman conquest, parish censuses show how serfs whose parents have Arabic names take Greek names; a testament to the dominant "Greek Christian" culture in spite of the new Norman ruling class and the almost entirely Arab administrative bureaucracy, and more importantly, the key role of Greek Churchmen in converting the Muslim population to Christianity. John Julius Norwich, in his The Normans in Sicily puts it this way: “Norman and Lombard, Greek and Saracen, Italian and Jew – Sicily had proved that for as long as they enjoyed an enlightened and impartial government, they could happily coexist; they could not coalesce.” Greek Orthodox inhabitants of Sicily could be expected to be fluent in Arabic, and some Muslims could be expected to be native Greek speakers. However, as with Arabic, Greek language and culture was also rapidly replaced in favor of Latin language and culture in Sicily and indeed all of Southern Italy. If you'd like to learn more, John Julius Norwich wrote a very extensive history of Sicily in two works, one chronicling the Norman Conquest and another on the actual history of the kingdom. Both works have recently been published together as a single work.

What your friend might be confused about is the status of the Italian Language in Southern Italy. Italian, as a language, only really existed in intellectual circles prior to universal education being introduced at the end of the nineteenth century; most Italians spoke their local dialect. In fact, Modern Italian is a construction by a clique of nineteenth-century intellectuals, who "modernized" the language of medieval Tuscan poets. Many people claim that the Tuscan dialect is Italian; but that is a common misconception, even held by some Italians. In reality, although there is a century-long history of intellectual debate on what constitutes "Real Italian," what is considered "correct" Italian nowadays was constructed by a clique of Piedmonto-Lombard poets and intellectuals who influenced public school curriculums at the end of the nineteenth century.

90% of Southern Italians were illiterate at the time of Unification (Negri-Zamagni, Vera. “La situazione economica e sociale nel meridione negli anni dell’unificazione: Una rivisitazione.” Meridiana, 2012) and these people would certainly have been only able to speak their local dialect. However, in all of Italy people's first language was their dialect at the time of unification. Even in Piedmont, the most advanced region at the time of unification, 30% of inhabitants were still illiterate, and use of the local dialect continued to be widespread. King Victor Emmanuel II himself had learned Piedmontese as his first language, and had been tutored in French; he never even learned to speak Italian! So although it's probable that some elements of southern dialects are borrowed from Greek and use of "Standard" Italian was very nearly absent, all Italians preferred speaking their local dialects.