Franks Kingdom
This is extremely doubtful. Where would the soldiers and the supplies to hold Italy come from? Look up the Sicilian Caliphates, they lasted only about a 100 years. Islam at this time is not a unity, and it gets less and less united the further you move away from the mid-east. There is no Islamic High Command planning and executing everything from Baghdad. The caliphs there couldn't give a damn what happens to individual caliphs in Spain or Italy, only the Ottomans, a united dynasty with geopolitical ambitions, had a vested interest in jockying their empire into the position of a unified front against a perceived christian unity. Christianity isn't a unity either, I find it simplified into the extreme to think of a united christian respons. The battle of Tours was ultimatively a Frankish problem, not an existential threat to the christian religion as a concept.
This is beginning to move from an informed discussion of the possible extent of the Arab conquest into "Blood Crazed Muslims Conquer the World, Pious Christians Drive Back Muslim Hordes Out of the Goodness of Their Hearts"-wank.
Given some of the previous posts, my position will summarize the facts and sociological and historical processes in which is based.
... An Islamic victory at Tours, possible, would not allow an indefinite expansion in Europe, the way it was imagined by Gibbon.
The victory did not save the Western Christianity of his disappearance, but it is certain that was the salvation for Christianity Iberian, Basque and Aquitaine.
Prestige and legitimate to Heristal, allowing the formation of a new German-'Romano'Cristiano Empire (Carolingian) with a Franco proclaimed and anointed by the Pope.
Which would expand the boundaries of his empire and with them Christendom, exterminating the Avars and the Saxons subjecting them and to become them by force of arms to Christianity.
What it would allow an Arab victory, it's politically controlling the region south of the River Loire to the Alps, threatening the Padana plain. Their permanent and undisputed possession of the richest and Romanized regions of Gaul, all Septimania, Provence. with the collaboration and the political support of its inhabitants, Goths, Gallic-Roman and Jewish. elapsed enough time to be initiated and completed a process of an gradual acculturation but inevitable. Time in which the Gallo-Roman culture will adopt the traits of Islamic culture, culminating in total acculturation of the inhabitants of Aquitaine on a few generations, as happened in Hispania.
Based on what happened in Otl, in regions that were conquered and remained in power of Islam; it is proven generally that the teaching of Religion or exchange of cultural features ranging from a more stronger culture circumstantially to "another weaker'' culture, it's seems that just as this process happened with Islam in Hispania and much of the regions that are Islamic in Otl, this could occur in Aquitaine under a permanent Islamic rule without too many Conflicts.
However, was observed that most transculturations were very conflicting, especially for members of culture 'receiving' less adabtables or more resistant to the new sociocultural context.
Remember, that Otl, most of the inhabitants and rulers of the Regions in Muslim power in Aquitaine, despite their religious differences between them and with respect to the Arabs, actively collaborated with the Caliphate forces at the time they taken their cities and later in his defense when the cities were attacked by the Franks.
It was very similar to the ethnic and cultural composition of Aquitaine to that in the cities of Hispania.
The expansion despite their defeat in Otl, continuous, but, to the East: take Arles in 735 and 737 take hold of Avignon, reaching
in raids the city of Lyon;
would not be until 759 (Otl), when Narbonne was taken by the king of the Franks Pepin's son Charles, who would fall Califal the last bastion of power in the region.
A victory would have prevented the Arab Islamic state isolation and stabilization as Emirate, behind the Pyrenees.
Being created a defensive mark, against the Franks, centered on the city of Burdeaux and the city of Arles, at the mouth of the river Rhone and Avignon , this region, at least partially Islamized and valleys Pyrenees and the Bay of Biscay ports in Califal power (and its successor the emirate), the Cantabrian ports and their hinterlands would be controlled.
Because of this the Realms of Astures and Basque mountaineers, could not evolve from the stage of isolated bands of rebels and 'bandits.
Referring to the Heristal (Carolingian), the victory gave them immense prestige, power and legitimate ambitions of convertrse in the unique power to govern Jure and De Facto in the Merovingian Frankish kingdom.
Based on his legacy his son intervened against the Lombards in favor of the Papacy, obtaining its legitimation as the monarch of the Kingdom Jure by the Pope, thus legitimizing the usurpation of his family to the throne.
Without the victory, his heirs had stage too busy trying to hold together the Kingdom and fighting rebellions of the conquered peoples or vassals: Frisians, Bavarians and Allamanes. Their children would have to continue to fight against their rivals Franks in a new civil war, as Carlos struggle and triumph 714-18 -724.
The faction that succeed even if it somehow prevent the Kingdom was divided again in the kingdoms of Austrasia and Neustria (Neustrasia), there would have neither the means nor the desire to engage in something other than the affairs of his kingdom, to achieve consolidate power, then the Monarch or monarchs Franks come back, as was their traditional custom, to divide his possessions among his sons; weakening again the Kingdom.
Situation that could lead quickly into another civil war, if any minor or illegitimate son (as was the case of Charles Martel), to rebel against royal authority and his heritage of territories of the Kingdom among his other heirs.
Perhaps allied to a Caliphal vassal or being exiled in Al Andaluz , obtained the support of emirate in their struggle for the throne in exchange for parts of the territory and / or alliance, conduct that would not be unusual for the Monarchs and aristocrats of the Kingdom Frank; with the Kingdom in this situation, weak, without unity of action, with its leaders divided and facing internal and external struggles. An Emirate stable in the north of the Pyrenees and at least part of the population of Languedoc and Basque, in the process of converting to Islam ,owned it human resources, wealthy of both regions and all Hispania.
Without inner border monitor and maintain causing the war and emigration depopulation of large regions in Iberia, these resources would be much higher; like the size and strength of their armies and navy.
Finally, even without territorial expansion, its merchants, protected by the power and prestige of the new and strengthened AL Andalus, and commercial enclaves propagate Islam and become the nuclei of small autochthonous Islamic communities.
Its economic role in Gaul (at least) would overcome and probably would replace the compliment by the Jewish community in the Burgh and the economy of the Kingdom Frank.