Well, they kind of did establish a reunited Europe. From ca. 780-843, there was a political unity from below the Pyrenees to the North Sea and from southern Italy to the English Channel (see map). But the ideological idea of empire -- the idea that all of Charlemagne's conquests were still a greater "Francia," that could (and should) someday be reunited -- persisted for much longer, into the 11th century at least.

The issue, as I suggest in my comment above, is one of definitions. What do we mean by empire? We tend to think of them as administratively coherent, with defined boundaries tied to geography. But that's a 19th-century definition, not a pre-modern one. Imperium (which we translate today as "empire") meant something more like "sphere of influence," so where a ruler exerted *imperium" could actually vary across his reign based on how people at the fringes of his control reacted to power flowing out from the center. What united the Carolingian empire, what kept the idea alive for several centuries even after Charlemagne's and Louis the Pious' death, was a willing adherence to an idea of ancestry -- to the idea that the aristocracy across almost all of Europe shared a common past that derived from the Golden Age of Charlemagne's reign.

tl;dr - yes, and they were successful in recreating empire (as long as you contextualize your definition of "empire").

The presupposition seems to be in this question that the Roman Empire united Europe. But the Roman Empire was primarily a unification of the Mediterranean area. Rome sure was invested in Britain and Gaul for mining, but depended on Sicily and Egypt for food. The power base of the Franks: the Low Countries, Germany and Northern France, were never more than a peripheral part of the Roman Empire.

After the collapse of the Western Roman Empire, the Eastern or Byzantine Empire continued to exist for many centuries, and indeed in the sixth century started to even expand westward, into Italy and Tunisia. Charlemagne's claim of empire confirmed the failure or end of that westward expansion. But the Carolingian empire or its successor the Holy Roman Empire never got into the position to seriously challenge the Byzantine dominance of Eastern Europe. Charlemagne did make inroads in Spain of course, but not in Britain. And as haimoofauxerre explains, later generations only even dreamed of reuniting Charlemagne's empire itself, never mind expanding it even further to the reaches of the Roman Empire. To imagine that it might have done so is a vast difference from its actual reach and priorities.

tl;dr - no, because the Carolingian Empire never had any influence over Eastern Europe, Britain or even Southern Spain.

Let me address the question from the point of view of: could the Carolingians have re-established a centralized state, such as existed in the West prior to the fifth century, throughout the former territory of the old Roman Empire? As opposed to a sort of theoretical hegemony, which haimoofauxerre very rightly points out was in fact established for a long time.

First off, let me say that the Carolingians appear to have tried. Charles didn't seem satisfied with just being the warlord over his domains. He tried to actually rule them, creating laws (the capitularies) and sending out officials (the missi) to enforce them. It didn't last. Why? Basically, because there was no money to pay for them. Charlemagne and his successors had to start granting fiefs to compensate his people, which turned the royal officals (including counts and dukes) into local hereditary nobles, leading to the medieval civilization we all know and love.

Why was there no money? A long, complicated question that's sure to elicit spirited debate. One group adheres to the Pirenne theory, which essentially boils down to this: the West was a mess, but redeemable, until the Muslims conquered Africa and the Near East and achieved dominance over the Mediterranean. The Mediterranean was the trade center of the Classical world, so its loss meant trade ground to a halt, taking money with it. Another theory -- once pooh poohed, but getting respect again -- says that it was the Barbarian invasions of the Fifth Century that done it. The Barbarians weren't out to destroy the Classical world, but they ended up doing so. The various groups each took their own piece of the West, cutting off its revenue from the rest, which meant the Romans couldn't afford to pay the legions and the navy, which caused the relatively secure economic sphere they'd controlled to collapse. It became too expensive to export anything, so large scale manufacture and agriculture also collapsed, further diminishing revenue, and a once sort-of united world became a bunch of petty principalities.

Whatever the cause, trade hadn't recovered by the eighth century, so there wasn't a lot of money out there, which meant a central government couldn't tax much (medieval rulers relied on the revenues of their own personal domains), which meant that there couldn't be the kind of infrastructure that would be required to maintain a Roman-type empire.

So, in that sense, they couldn't have done it.

Sources: Charlemagne: the Formation of a European Identity by the charming Rosamund McKitterick for a nice overview of Charlemagne's attempts to rule.

Feudal Society by Marc Bloch is a classic description of the collapse of post-Charlemagne control into feudalism

Henri Pirenne's Mohammed and Charlemagne is his fullest description of his theory for the end of Classical civilization in the West

The Fall of Rome and the End of Civilization by Bryan Ward-Perkins is one nicely readable Barbarian-theory revival title. (And fighting words, I'd wager.)

I seem to recall something from one of my history classes that stated that Charlemagne at one time proposed marriage to a ruling Empress in Constantinople, but there was a revolt and the new Emperor that took over was not interested in uniting the two empires. Don't have a source.