This is not the Roman Empire of the Last Emperor's days; and seeing as this has not been stressed yet; conquering Constantinople does not conquer Rome. Rome the city fell and was sacked many times, Rome the state did not. The same applies to Constantinople. The city may fall, Rome will (and indeed, did) persevere. The rest of the Roman Empire is still there, and presumably now enraged that the Mohammedans just sacked their capital. The Roman Navy isn't going to mass-defect to sworn enemies, the Roman nobility isn't going to stop fighting either. At best they just pissed off a medieval superpower, well done guys, bravo.
I disagree.
Constantinople was more important to the Roman Empire in its Byzantine form than old Rome was to the Empire in its earlier periods. Constantinople really was, in many ways, the heart of the empire, a city which only grew in importance, in contrast to Rome, which steadily declined in importance to the governing and maintenance of the empire during the Republic, Principate, and Dominate. A lot of that had to do with its much more strategic location.
In any event, Rome survived in the 5th century despite the fall of Rome and even Ravenna because there was plenty of empire left, an entire eastern half virtually untouched and even growing more prosperous. But where does a Byzantium of the 670s that has lost Constantinople fall back on? Initially, probably Thessalonica, but that's much less defensible than Constantinople, and with less resources in its hinterlands, and isolated by Slavic war bands... It has lost not only its Semitic realms but also Anatolia, its prime recruiting ground; the Balkans are largely overrun by Slavs and Avars. North Africa is about to be overrun by the Arabs, too. It retains Sicily and the islands but these are now highly vulnerable to Arab sea raiders. The Exarchate of Ravenna holds on, barely, against the Lombards, but that would hardly form much of a base for the Empire to survive as anything but a weak local successor state, like Trebizond was in the 14th-15th centuries.
They're magically going to conquer Europe now? After this miraculous victory the likes of which people will tell horror stories about for generations? The Merovingians may have more than one thing to say about that, as well as a substantial part of the Christian (and indeed, pagan!) kings and chieftains of Europe. No, it's not the Battle of Tours yet but that could very easily come to a head a generation early. In my opinion; No, you're not going to see Suddenly, Mohammedan Rome. No, you're not going to see it in Lombardy, either, nor anywhere else really as welcome to logistical woes the likes of which can't really be described because they had this problem with trying to quite literally conquer the entire world all at once and were stretched so thin as for it to be considered downright laughable, if the Mohammedan Conquests didn't happen because of downright incredible luck more than any sort of political or military savvy, they'd be regarded as fantasy today at best. Is it plausible for the Mohammedans to have won in 678? Perhaps it is. Is it plausible for them to then go on, and conquer most of an entire continent while conveniently ignoring all the already extant problems they faced now compounded by even more? No.
If the Umayyads overrun Constantinople, most of Greece and the Balkans are almost certainly forfeit within a few generations; and no longer standing in the way of Arab attentions on Italy, which faced plenty of attacks in the 8th-10th centuries as it was. I think the odds are against keeping Rome; the Papacy likely relocates to Ravenna, which is more defensible; and if the Franks still rise to power here, the popes might even move to Aachen. There's a limit to what the Arabs can reach, but at the least, I think Italy, the south of France, and the middle Danube become war zones, and Frankish resources more heavily diverted to dealing with the new threat from the south. This might make campaigns against the Saxons and beyond the Elbe infeasible.