The name Italian is used for the Romance language, so you probably mean the Romance languages came from some Italic language(s), which coexisted with Latin?
We know quite a bit about the non-Latin Italic languages, which include Umbrian, Faliscan, Oscan and some others. They were spoken in Italy and were related to Latin. No doubt they had some influence upon Latin, especially before the classical era, but, from what we know, Latin resembles the Romance languages more closely than do the other Italic languages. So it seems extremely unlikely that e.g. early Italian should have developed from e.g. Oscan.
We also know quite a bit about Vulgar Latin, 'Latin of the People', which was spoken by the common people in Antiquity. It is somewhat different from classical/literary Latin, but it's quite close. Information on Vulgar Latin is mostly from source such as graffiti, quotations, and comedy, which often emulated the Latin of the common people. The Romance languages are thought to have developed from Vulgar Latin, and we have quite a bit of information about intermediate stages, which match what one would expect.
In case you meant "other, non-Italic languages spoken in Italy" by your Old Italian, then it would seem even less likely, for they would resemble the Romance languages far less than do the Italic languages. The major non-Italic languages spoken around Italy by the time the Romance languages took shape, in the early Middle Ages, were Latin, Greek, Germanic languages, and probably some scarce remnants of Etruscan. Greek is from the Greek branch of the Indo-European family of languages; Germanic is of the Germanic branch; and Latin is of the Italic branch, like the Romance languages. Resemblance is closest within the same branch. From the little we know about Etruscan, it does not resemble Latin at all, for it is not just of a different branch, but of an entirely different (unknown) family.
I would therefore not take this Cortez very seriously, unless by Old Italian he meant simply "Vulgar Latin" or some intermediate stage between Vulgar Latin and Mediaeval Italian. In that case, I would want to know which exact stage this would be; but, truly, I don't think it makes sense to think that French developed from a local, non-standard Italian variant of Latin in the early Middle Ages, when there had existed old and established Latin-speaking cities in France for half a millennium. Why would they switch to a strange variant of the same language (Latin) from a thousand kilometres away (from Italy) only at that time?