Algonquian languages are not organized along at all similar lines as any language I know from Europe. At all. Definitely not anything Indo-European, and probably not any pre-Indo-European substrate languages of the Mediterranean.
If I was keen on proving the author's theory, I would start analyzing proto-Algonquian grammar and see if there is any solid, regular connection with linguistic substrata in Europe (Basque, Etruscan, pre-Greek, etc.). But it's pretty doubtful. I actually came to Algonquian grammar after having studied Latin, Greek, Etruscan, Osco-Umbrian etc. It's a pretty stark difference...Mediterranean languages are typically inflected, whereas Algonquian, like many American languages, is agglutinative.
To make a long story short, I wouldn't put a whole lot of stock in this hypothesis the author lays out.
I was just about to ping you to this. Not surprisingly, you’re already here.
The relationships between these languages are not readily established by looking at vocabulary (since your typical agglutinative language readily absorbs words from any source). At the same time there are relationships somewhere. DNA studies suggest they are in the Western European Ice Age refugia in Iberia!
Once you know that there's a DNA sequence peculiar to the first group(s) to leave the refugia at the time of the big meltdown (14,000 years ago) it's a simple matter to look for others. That has been done ~ they are Berber, Fulbe, Sa'ami, Cherokee, Iroquois, Chippewa/Ojibway, and Yakuts.
Follow up with DNA should certainly help clarify any Mediterranean connection. On a related subject, does anyone know about reports by early explorers that the Mandan Indians spoke Welsh, or something with Welsh words? I think that the exact statement was that a Weshman in the expedition could understand/communicate with them. I seem to recall they were quickly wiped out by smallpox.