So late to the discussion, so much to read!
I think that the Roman Empire would've eventually fell because all empires fall, especially the classical ones. Best case scenario for a modern day Roman empire would be something like the Chinese have where you've got periods of unification and of warring states.
Of course the Roman Empire could survive. Very little in history is inevitable, and the rise and fall of empires is even less so. Of course there will be low tides and high tides over the centuries, there will be territorial losses and gains, but the fall of the Roman Empire was never inevitable. It could survive solely in the east, it could survive with largely the same territory it had at its height, or it could survive in any number of fashions in between.
I have to agree with Sly here, we only say it is inevitable, because IOTL no Empire has survived. But the Byzantines survived for an exemplary period of time compared to other Empires - and I'm sure at the time, nobody expected it to fall.
Carthage, Alexandria, Tarraco (in Spain), Milan. Depending on how far east the empire stretches, Antioch.
I quite like the idea of Carthage, sweet irony indeed
Tarraco is also quite interesting.
Carthage does work as a fantastic captial for a Roman Exarchate, which for a fallback is pretty crucial. (The idea of a collapsed Empire, and the Roman Emperor becoming more symbolic whilst the Exarch has the real power is quite an intriguing one.)
Tarraco could work too, but Iberia isn't nearly as defensible as Africa. It could work if Iberian and Africa are secure.
You might find
this to be of interest.
Omg Yiis, this is awesome. I may or may not have to use that extensively in something I'm writing.
But fundamentally, to the OP - yes, the Roman Empire could survive to this day, as a continual institution. There are simply a number of challenges the Romans have to overcome, permenantly.
1) Stable succession
2) High quality, robust governance
3) Military struggles, these probably highlight the best starting points for PoD's that make the Empire survive longer - which preludes permenance.
So my choice PoDs?
1) Strong Caesarean Dynastic Rule
2) Theodosius creates a stable/sensible succession by living longer
3) Victory at Yarmouk
I think 3 is probably my preference, being later on - but the ERE, under a strong Heraclius, bolstered by victory at Yarmouk, could very well recover from the Persian Wars faster than Persia, and Arabia is beaten back for the moment.
Depending on the response after the victory (is it wise to take Mesopotamia? Can the Romans take Arabia and prevent them being a threat?), we can probably see the Empire survive. A Roman Arabia is largely a source of desert cavalry that would exist as a convenient fast cavalr force against Persian invasions (and potentially offensives against Persia, good idea or bad), and some more influence in the Indian Ocean. A Roman Mesopotamia is a very wealthy addition to the Empire, but fundamentally relies on whether it is defensible from the Iranian Plateau - which might take most of the wealth of Mesopotamia. But with both of those in Roman hands? It frees up forces to go west. This is where control of Arabia really shines - Bedouin cavalry, brought in, using desert warfare in the Sahara can really help the Romans, and makes Africa more secure, and could very well enable the Romans to take control of the various oases that enable people to invade Africa. Taking those and fortifying them, is as good as building a wall the length of the Rhine, making Africa more secure. (Sidenote : this could make it secure enough for the fantastic idea of a canal to expand the Chott el Djerid in Tunisia)
With the Sahara, Mesopotamia, and the Balkans as secure(ish) buffer zones, the core Empire is safe - and in a fantastic economic position. I have every faith that with good campaigns in Iberia and Italia, after allowing their demographics to recover, and a bit of cunning in Persia, Arabia, and Frankia - that the Romans could rebuild at the least a "Mare Nostrum". After that it is good governance, and it is early enough to ensure that being "Roman" is the main cultural identity.
Throw in absurd wealth from potentially discovering the New World? Then the Roman Empire can afford better defenses, and potentially expansion to shorter European borders.
TL;DR Yarmouk has a big turning point, if the Romans can win it, and turn the next century into a good one, then they have every chance of making the next milennia a stable, or expansionary period, rather than a steady decline. A Roman Empire of that size in the 1600's/1700's is going to exist till today.