The Battle of Tours is widely touted, along with Salamis, as a battle that saved western civilization.

This really is not the case. I've phrased it this way on this subreddit before, but the Umayyad war machine was grinding to a halt by the time armies crossed the Pyrenees. Supply lines were massive, especially for an empire stretching from Transoxiana to the Iberian Peninsula. Also, keep in mind that this Arab army was most likely a raiding force out pillaging Aquitaine rather than a force bent on conquest.

Coupled with that, a few years following the Battle of Tours there was an Umayyad disaster in North Africa. The Berber Revolt of 740 crippled imperial authority in North Africa and several independent Islamic Berber states rose. The Umayyads suffered a humiliating defeat and couldn't focus on foreign conquest.

So, realistically, had the Umayyads emerged victorious from Tours, they probably wouldn't have been able to follow up the victory.

Well the Umayyad takeover of Spain was in many ways more about opportunism rather than outright conquest.

Their forays into Spain could in many ways be seen as an extension of the Arab raiding of Anatolia (for prestige and booty), and only settlement over a longer span of time once the frontiers had moved past the old raiding areas and became stable for farmers to move in.

It just so happened to be the good luck (to the muslims) that the Visigoths had just survived a nasty civil war which had left an usurper as king with questionable legitimacy, with that king being killed in battle with the Arab and Berber army. With King Roderic's already shaky legitimacy, his death brought down the whole edifice of centralized resistance.

Keep in mind the Iberian peninsula, was already already prone to independence amongst its regions (take for example, the long survival of basque culture from even before Roman conquest) because of its particular geography. The Muslim army offered good terms to many of the remaining Visigothic leadership, and many regional governors cut deals with the Arab leadership in exchange for autonomy.

It was only after more than a hundred years that Romano-Visigothic culture switched over to Islamic, for reasons similar to Egypt and the Levant: Islamic religion required conversion in order for people to participate in its politics. Thus despite not being a forceful process, it was an attractive one nonetheless.

Why all this background on Visigothic-Umayyad Spain? Because this was and would be the situation in France as well regarding Tours.

It's interesting the precedents in Charles Martel's era that are similar with the last king of Visigothic Spain. You have a contested leadership conflict. You have autonomous outlying regions of the kingdom that the central leadership is trying to recentralize by force. You have opportunistic raiding by the Arabs, that sometimes changes into occupation with a successful seizure of a fortified town. In the event the Arabs can't seize a key town, they call it a successful raid on the countryside and then go home.

So, supposing the Merovingians lose Tours. What would happen?

In my opinion, given the precedents, much would depend upon the survival of Charles Martel personally, and to a lesser degree the Merovingian "king" Theuderic IV. The death, be it physical or political, of Martel (like say a situation where he completely lost legitimacy by fleeing the battlefield like a coward), and say the subsequent capture of Theuderic would lead, much like in Visigothic Spain, to a collapse of centralized resistance. Martel already had to expend a lot of energy reconquering fairly autonomous lands in his own core provinces in the 730s. With him gone, those regional leaders might decide, like the Visigothic governors, to recognize Arab authority in exchange for autonomy.

Now, supposing Martel survives, loses the battle, but puts on a good show and keeps together a significant degree of his forces, then the particular concerns of supply chains come into play. If Martel's forces have lost but are intact, though he can't face the Arab army in pitched battle, he can harass them while on friendly terrain, thus keeping the Arabs from simply walking into key towns/fortifications, or negotiating from a position of power with the controllers of those locations.

And likely what'll happen, is the Arabs will declare victory, wander back to their home base, and continue raiding next year. If an opportunity comes up again to defeat the forces of whoever the Frankish leader is again, they'll likely take it and see where it goes. Once again, opportunism, rather than outright conquest.

Keep in mind too, should Martel lose Tours, even if he keeps his forces together, he's also lost a significant bit of his legitimacy, so the problems of autonomous former provinces pops its head up again. But the difference would be, without the Arab army sticking around unopposed, it'll be a Frankish cultural fight for autonomy, rather than local autonomy under Arab suzerainty.

tl;dr - Did the Umayyads have a reasonable hope of taking over Francia? Sure. But it wouldn't be subjugation. It would be voluntary opportunism on both sides, both of the desires of autonomy from local frankish aristocracy, and the success of arab armies in a pitched battle.

First I would like to mention that there is dispute as to the year in which the battle took place. Given that the date is in question this you should consider that many critical details around lines of supply, training, arms, politics are simply unknown.

The date of the Battle was thought to have happened in 732 but White and others believe it happened in 733. Wikipedia which sums it up far better than I can. 27^ Oman, 1960, p. 167, gives the traditional date of October 10, 732. Lynn White, Jr., Medieval Technology and Social Change, 1962, citing M. Baudot, 1955, goes with October 17, 733. Roger Collins, The Arab Conquest of Spain, 1989, concludes "late (October?) 733" based on the "likely" appointment date of the successor of Abdul Rahman, who was killed in the battle. See White, p. 3, note 3, and Collins, pp. 90-91.

The referenced Lynn White's "Medieval Technology and Social Change" touches on the Battle of Tours which also is called The Battle of Poitiers because of the introduction of the stirrup. Full-foot stirrups began being widely used right in AD 732. Also Charles Martel takes church property in 732.

If the battle happened in 733, then Martel has better technology via stirrups and resources from church property to oppose the invasion.

Dr. John Leinhard, an engineering prof who teaches the history of technology, from the University of Houston goes over the battle and stirrups http://www.uh.edu/engines/epi476.htm