Okay, Jewish culture dominant in the Med.
Well, if you look back at your history books, you can see that this is technically possible. Arab culture has spread through most of the Mediterranean, and an assortment of Germanic and Latin culture groups were scattered. Judging by how the Middle East managed to stay unified for a while and spread its culture and religion, I'll start there.
The obvious path is to make the Caliphate Jewish. But how?
Okay, the Caliphate under the Abbasids was going through a very rough spot, starting in about 800, and continuing for the next couple of centuries.
This is a good time to implement an invasion by the Khazars. Most of you should know about these guys, but for those who don't, I'll explain. The Khazars were a Turkic Tribe that lived in the southwest steppe and northenr Caucasus, and are also known as the only Jewish state other than Israel (or its many reincarnations).
At this time the Khazars aren't Jews... yet. They lost (kind of) a war to the Caliphate in the 750s, and probably weren't happy about it. So tweak things a little, and the Khazar hords tromp south again during the Caliph's rough patch, sensing weakness.
As the Caliph's troops are in bad condition and elsewhere suppressing various uprising, the Khazars are given a free passage south through the Middle East, reaching Baghdad quickly.
Let's say that the Khazars take a huge amount of territory in the northern part of the Caliphate, and put one of their royal members on the Caliph's throne. In Anatolia, the Byzantines step the Arabs down a few pegs, expanding a lot.
So we've got a Caliphate that's been weakened, and shortly its North African possessions are breaking off, as are many fringe territory. In the confusion and chaos that follows, various Jewish communities and families emerge in a position of power, controlling trade, production, tax collecting, and the general beuracracy of the Caliphate's system. So now we've got a Caliphate that's relying on Jews to keep it toghether.
I'm inclined to think that the Khazars will choose to convert as OTL, but that may not be the case. Let's assume it is, though, and the Khazars (and the Caliph) convert to Judaism.
For the next few years, various factions turn to Judaism, and Jews are persecuted throughout the Caliphate. For a brief period, the Jewish population blossoms, erupting into areas wiped clean of Muslims in war.
Now, the Khazar control over the Caliphate won't last very long, and when it does, the Jews of Arabia are in trouble. So let's say that huge civil wars erupt when the Jews are betrayed, and many Jews choose to leave in the new Fatamid Empire in North Africa, which was tolerant of Jews, or Europe.
Jews are now widespread, and growing. The Caliphate falls apart a couple hundred years earlier, and a Jewish state takes hold in Palestine, which flourishes. The crusades never happen, but the Jews are now seen much better in the eyes of Christians just because there are so many of them.
The North African Jews start there own cities and towns, outnumbering the Muslims in a few places. Assuming that what happened in Khazaria OTL will happen here, North Africa under the Fatimids is officially Jewified. With help from the Fatimids, *Israel (probably wouldn't be called that) conquers many of its neighboring Arab states, and Jewish influence is spread through the Byzantine Empire and the Middle East.
Then the Mongols arrive, and things get tricky. Assuming that Khazaria has survived because of its extensive network of allies and friends, they're the first to be attacked. The Mongols suffer several setbacks, but the Khazars fall. Expect a mass migration south, either down the Caucasus, or west and then south into the Balkans were the friendly Byzantine Empire dwells.
The Arabs are definitely not in a good position, and serve as a buffer between the Mongols and the Jewish states.
So once the Mongols splinter and fall like OTL, the Jews in North Africa and *Israel are relatively unharmed, and it's up to them to pick up the pieces of broken Europe. Let's say it starts in Iberia, where the dying Muslim caliphate there falls to waves of Jewish Fatimide invaders, and added to the Fatimid King's gains. The Byzantines would probably get more of Anatolia (possibly all of it) and more of the Balkans. Within a few generations, they're sufficiently Jewified. The influence of *Israel, Byzantines, and the Fatamids is enough to put European Jews in a good position.
From their bases in Sicily, the Jews trade and settle in Italy, which is being ravaged by the Plague. While equally affected by the plague, the Jews have the advantage number-wise, and much of Italy is now Jewified. Eventually, this happens to the rest of the Mediterranean basin.
Where does it go from there? I can't honestly say. But now we've got the foundations for a Jewish Europe, I have to elaborate on a few points.
First, much of the northern Middle East and the Balkans are Turkic from the Khazar migrations.
Second, Arab traditions would be reabsorbed back into Judaism over time
Third, even if Judaism as a religion doesn't stay around, it'll still have enough influence to reflect on the culture of most of the Med for centuries to come.
Note: It's late, so I probably got a lot of this wrong, so don't chew my arse off because of it.