Counter-point: One of the prime motivations for European colonialism was competition. Without any home-continent competition, there is relatively little "push" motivation to explore. The desire to seek new sourc es of spices still exists, but again without political boundaries in place, there would be little reason to find a navigable route west to the known spice producing areas.
I think a major reason was that the Islamic world (specifically the Ottoman Empire) was "in the way" to the East Indies, and going around Africa was bothersome. It's not like European powers just wanted to one-up each other: Europe wanted direct access to the far east and its precious commodities. The whole impulse to go west (around the globe) was prompted chiefly by that desire for direct access. It failed, obviously, but America proved to be highly valuable in its own right.
In short: if there's a major power in the way of the far east, Europeans going west still makes sense. If there's no major power in the way, I'd conversely expect the Empire to indeed ignore the Americas to a great extent, but expand further east instead. (Permanently retaking Constantinople and the Levant, for instance-- probably trying to get Anatolia and Egypt/Libya as well...) Either way, I re-iterate that I'd expect this Empire to be more expansive in at least one direction.
Of course, a deliberately less expansionist Carolingian Empire ("Europe as China", as
@B_Munro succinctly phrased it) would be fascinating in its own right. Problem is, for that to happen, I really think you'd need to develop a mindset averse to oceanic exploration
before Europe reaches the Americas. If you do that, you get a nicely isolated Europe, forming its own cultural sphere. Indeed, "a culture pretending to be a country/empire", just as China has been described upon occasion. But once the Europeans have decided that starting colonisation in the Americas is worth it, the vast value of the New World will almost certainly motivate them to go "all in". Without competition, it may be a much slower process than in OTL, and more Native client states might be set up instead of full settler colonialism, but since this is a modern-day map, I'd really expect the Empire to control far, far more. (In fact, to control -- if in large part indirectly -- all of the Americas.)
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Seems to me that it's already a a massive colonial power and most of the colonization was in Europe.
I'm not so sure. While I find it interesting to consider Christianisation and cultural change imposed on the direct neighbours as, indeed, a form of colonialism, I don't think that after a few hundred years, the various European and Christian peoples of the Empire will consider themselves any less part of the whole than, say, the Saxons of 1200 AD did in OTL. (Remember, said Saxons were a thorn in Charlemagne's side as late as 800 AD. Just a few centuries later... well-integrated within Christendom and the imperial structure of the HRE.)
There's no reason to assume that this massive state was or is massively stable between its various constituent parts, nor that colonization is necessarily a European thing. Perhaps it has a West African rival, or is more focused on the Indian Ocean, or is unable to extend its reach very far beyond its borders (this is, after all, the entity that in our universe became the HRE).
We see a modern-day map here. If this empire isn't essentially stable, I refuse to believe the basic premise that it survived from 800 AD until the present day. That doesn't happen when Empires are fundamentally unstable. This Empire successfully reconquered significant parts of North Africa and did manage to colonise the American eastern seaboard to the extent that it is (apparently) directly part of the imperial structure. (Maybe that explains everything? Has
@Mario just decided to only include direct parts of the 'metropole', and just not showing any less-directly-included colonies?)
Regarding rivals... I don't see a West frican power rising up, with a Charlemagne-era POD, and becoming powerful enough to go toe-to-toe with a united European Empire. If we go with multiple PODs (and include an earlier one in West Africa), that becomes viable. But from what we see here, a West african power competing with the Carolingian Empire seems about as likely as the OTL Aztecs competing with them. That is:
not very.
If Europe is more focused on the Indian Ocean, I fully agree that interest in the Americas would be decreased. Problem is -- as I pointed out above -- that this requires access to the Indian Ocean. Which almost by definition demands that there is no powerful Islamic rival to the direct east. Which in turn begs the question: why don't the Carolingians own more land in the east?
More than anything, it seems to me that they found the New World and decided there wasn't much of interest and that colonization was slow and plodding, unsupported by the ruling class (which would make sense if they were worried about constituent states/provinces/kingdoms/what-have-you trying to strengthen their position within the Empire through colonization) and subject to the Empire enforcing dominion only as far as they saw necessary to preserve the balance in Europe.
The new world seeming uninteresting strikes me as highly unlikely, considering the sheer wealth it offers. If there is free access to the commodities of the far east, sure, but again, then I'd expect the Empire to have expanded futher east. (I mean... if you're going all in on the Indian Ocean, you want to own Egypt and ideally Mesopotamia.)
Fundamental instability in the Empire, I stress again, is hardly likely if we're expected to believe it lasted -- at the size shown -- for some 1200 years or so. And furthermore, even if there have been bouts of troubles, colonies have historically been dumping grounds for excess troublemakes (and criminals, and poor). If you expect there to be a rather feudal elite running affairs, a growing population is nothing they want. They'd be the first to
support colonialism, as a way to get rid of everyone they don't want in Europe!
(Again, I offer the suggestion that all or most of the Americas were in fact colonised, but a bunch of colonies, populated mainly be such 'exiled masses', have since revolted and broken away.)
Assuming this entity would conquer everything by virtue of Britain, France, Spain, and Portugal not existing as independent entities is like looking at a map of the Spanish Empire in the year 1600 and saying "why didn't they just conquer all of Europe? Look at how much land they control!" The answer is that's not how foreign policy works, that's not how empires work, and more specifically, that's not how their foreign policy or empire worked.
This comparison is completely wrong, chiefly because Spain had lots of competitors in Europe, which reopeatedly united to fight against the power of (first) the mighty Habsburg Empire and (later) the Spanish crown and its vast colonial empire. This Carolingian empire, conversely, has no such rivals on the Atlantic. It is free to colonise the Americas without any real competition. It can use the vast,
vast wealth gained from this to increase its own wealth and power exponentially. Even if rather mismanaged, and largely ruled by satisfied aristocrats resting on their laurels, it should still be larger than what we see here. (Or at least, it must at one point have been larger-- as said, independence wars having happened in the past is a realistic option.)
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...man, this discussion grew far more in-depth than I initially intended or expected when posting that intial comment on
@Mario's map. I blame you far all of this, Mario! Now end this discussion, and let us wonder no longer: that is the true background of this scenario you have offered up?