Let us say that the events of 628 and the Byzantine-Persian war go as planned. The only difference is that Islam does not exist. Without the pressures of Muslim encroachment, the stresses they placed on Heraclius, and so on, what would happen to Byzantine Italy?
The Lombards are still around and so is the pope. What would happen there if Heraclius has his way with the Monothelites, which according to some people I've heard from before (namely BG) was actually pretty successful in mending the Monphysite-Melkite gap? How would it do with Catholicism? And how would this impact the Pope and the Lombards, especially with Italy in a more stable position without distractions from the Arabs?
I think what I've said and read is that the Monothelite compromise went down reasonably well with the previously intractable bishops of Egypt and Syria, who percieved that they'd dragged the Chalcedonian heretics back to the path of truth and light. Constantinople's Patriarchate was also solidly behind the compromise, and it was the monks of Palestine that opposed it so heavily.
In a no-Muhammad scenario, then I see the compromise doctrine continuing in force for some decades, in the way that Iconoclasm did. If an heretical Emperor won military victories, then his beliefs could generally hold on despite limited to no grassroots support and serious Papal disapproval, as with Leo III and Constantine V.
Without an Arab explosion in the 630s Heraclius is one of the greatest victors in Roman history, as opposed to perhaps having the most disastrous reign of all as IOTL. The fate of Monothelitism perhaps depends on who succeeds him: due to butterflies, it's possible that the short-reigned Constantine III might enjoy a decent length reign in the 640s and 650s. Constantine is likely, in my mind, to try to restore a measure of Roman control in the Balkans, as well as possibly sparring with a resurgent Iran in the East. Should his reign even be a minor "net positive" militarily, then his doctrines are likely to be percieved to have divine blessing: although should he suffer even minor setbacks, the converse is likely.
The Papacy in the seventh century is simply too weak and dependent upon Constantinople to do much to seriously protest: after all, in the OTL 650s Constans II was able to rule from the West and repeatedly crush Papal opposition. Emperors in this ATL are going to be able to behave with even more impunity than Constans II did IOTL.
Iconoclasm suggests to my mind that as doctrinal innovation depended upon a period of military supremacy. Should the Empire avoid a serious disaster in war (or the accession of a determindly Chalcedonian ruler), then Monothelitism is going to stay solid no matter what the Popes have to say about it.