It's entirely possible, however unlikely in the face of "1800s Manifest Destiny," that the Great Plains and 'Indian Country' could become a buffer state between the Romans and whoever their neighbors turn out to be.
The question is; with larger and more disease resistant natives, at least in the more western half of the continent, in Mēxihco and anywhere that the Central American trade routes go (which, IIRC, there is evidence for them going quite deep into both North and South America), will Manifest Destiny still be a thing?
A major supporting foundation of Manifest Destiny was the concept that the land was
empty, that natives were few and far between and that there was nothing standing in the way of colonization and expansion. Which was, broadly, somewhat true thanks to like 90% of the native population being wiped out by disease.
If the natives have already started developing immunities to European diseases thanks to earlier contact with them from Constantinople, then there will be a lot more of them and they will be a lot more organized by the time any ideas of Manifest Destiny roll around, and that's not taking into account the effect that knowledge of the
definitively Chosen City Of God being on the other side of the continent would have on people's attitudes towards colonization.
It's one thing to dismiss the
Very-Western Roman Empire as 'natives' and unimportant when you're only meeting slightly European-looking native people, it's quite another to do the same when you're staring at
the City of Constantinople that somehow fucking teleported onto the other side of the planet to escape Mehmed the Conqueror.
That kind of solid, definitive, provable miracle that you can literally point to and go 'what the fuck?' isn't something that can just be ignored unless you are a long way away from it and can pretend it doesn't exist.
Plus, given the Romans historical focus on assimilation rather than conquest and enslavement, combined with Saint Constantine's divine mandate towards the natives, I'd expect to see a lot of semi-allied native polities along the fringes of Roman territory: The whole thing with the Purépecha went so well that I can't see the Very-Western Roman Empire not trying that sort of thing again with other native groups, which could easily lead to a whole bunch of 'semi-Romanized' native polities dotting the middle of the continent as a
much more resilient and organized 'buffer' than OTL, which would have knock-on effects for any potential Manifest Destiny moves.
I'd expect to see that North America ends up split into three vague 'strips' with the Romans in the west, the Europeans in the east and a bunch of native confederacies and alliances with ties to the Romans in the middle. (No clue about how Canada might go.) The other thing that comes to mind is that IIRC there was a lot of wheeling and dealing with regards to 'ownership' of North America, with England and France doing the diplomacy and the politics out the wazoo while trading chunks of land around. That will be another part where the presence of larger, more organized, more technologically advanced and more resilient Native populations cause a lot of changes, not to mention Constantinople itself.
(Keep in mind; there were multiple native communities in the Great Plains area that were totally annihilated by disease
long before any European ever even set eyes upon the region, communities that we only know existed today from hearing of them through other native communities. And
literally the only thing we know about these peoples is that they
used to exist somewhere in this general area, and that they did not exist by the time any white people showed up. The Roman introduction of European diseases
might change that.)
Central and South America will go
very differently; amongst other things there is no chance at all that Moctezuma will just let Cortés saunter on in to Tenochtitlan in order to 'learn his weaknesses', which means no easy coup and sacking of the city followed by a hasty getaway laden down with gold. That alone will have a significant impact, as without ludicrous amounts of free gold to wave around I suspect that Cortés will find avoiding the consequences for his many crimes to be rather more difficult than it was in OTL. And that's not even getting into the knock-on effects that the presence of the Purépecha as an organized, partially Romanized nation will have on the area.
Might the Romans end up making Arcadia into a sort of North American equivalent of Peru in terms of country shape? Plenty of room to just expand up and down the Western Coast of North America, up to where Vancouver now is and down to what in TTL is the territory of their native allies. They're not going to see the English/Dutch for another 200 years at least so for now it's more the Spanish, Portuguese and French Empires that will be their problem.
They
will be looking inland, but at the very least early colonization efforts are probably going to be along the coast yes: Building the necessary roads for inland travel and colonization is a time consuming and expensive process, even for the archetypal road builders of Rome. Ships on the other hand are much cheaper in comparison and can travel anywhere along the coast at greater speed and carrying more cargo than land wagons, so the initial Roman expansion will probably be along the coast, and Roman territory in general is unlikely to get past the Rockies any time in the next couple of centuries. (The Rockies are a
serious barrier to conventional ~15th century travel technology, it is no coincidence that European colonization didn't get around those mountains until literally centuries later.)
That said, the natives
do trade with each other, so while direct Roman influence is unlikely to pierce the Rockies, Roman technologies and ideas may well spread across them and into native communities on the other side.
