Usually, when one learns about the Roman empire in school, it is in the context of a westernist narrative, where it is said to be (together with Greece) the cradle, or even prefiguring, modern European civilization. Now, I know that this modernization of Rome and Greece is, in part at least, a bourgeois myth, and that institutions like the Republic, Roman Law, or Democracy in Athens were in reality not as progressive as they are made to be in school curriculums; They were, in a way, re-interpreted and reappropiated to fit the revolutionary ideals of the Enlightenment, and to create a mythic past on which to base the new bourgeois societies. Or in Marx's words
"The tradition of all dead generations weighs like a nightmare on the brains of the living. And just as they seem to be occupied with revolutionizing themselves and things, creating something that did not exist before, precisely in such epochs of revolutionary crisis they anxiously conjure up the spirits of the past to their service, borrowing from them names, battle slogans, and costumes in order to present this new scene in world history in time-honored disguise and borrowed language. Thus Luther put on the mask of the Apostle Paul, the Revolution of 1789-1814 draped itself alternately in the guise of the Roman Republic and the Roman Empire [...] When we think about this conjuring up of the dead of world history, a salient difference reveals itself. Camille Desmoulins, Danton, Robespierre, St. Just, Napoleon, the heroes as well as the parties and the masses of the old French Revolution, performed the task of their time – that of unchaining and establishing modern bourgeois society – in Roman costumes and with Roman phrases. The first one destroyed the feudal foundation and cut off the feudal heads that had grown on it. The other created inside France the only conditions under which free competition could be developed, parceled-out land properly used, and the unfettered productive power of the nation employed; and beyond the French borders it swept away feudal institutions everywhere, to provide, as far as necessary, bourgeois society in France with an appropriate up-to-date environment on the European continent"
In this way, we've come to think of Rome as inextricably liked to modern Europe. Understandably, I was kind of surprised when I first learned that not only was the Western half of the Empire kind of a periphery to the more developed and urbanized East (Anatolia, Levant, Egypt among other regions), but that northwestern Europe (the place where the bourgeois revolutions first occurred) was itself peripheral to the core regions of the Western Empire (Italy, Northern Africa and Southern Spain). I know that this sort of makes sense when we think that the Empire was first and foremost a Mediterranean geopolitic construct, but it's still weird in a way, as most of those core regions are what we would now call the Islamic world. It's kind of ironic, considering all this Clash of Civilizations rhetoric that's so in fashion right now.
So, how western was the Roman Empire?
Edit: spelling, quote.
Well, just ask yourself: where do people have public baths?
As a more serious answer, Rome was a Mediterranean empire. This isn't quite what you wanted, but the majority of its population and economy was concentrated near the Mediterranean coast. It doesn't easily fit in with modern conceptions of East and West precisely because no other state has ever maintained control along the entire sea, and no other culture has so thoroughly pervaded the entire region. Rome was very much a historical oddity.
I do personally think the marginality of continental Europe to Rome is often overstated, but you are correct that the Islamic regions of the Eastern and Southern Mediterranean have every bit as much of a share in the classical heritage as Italy and Greece. Personally, I can't abide those who try to write the Muslims out of the story of Classical civilization.
I agree with /u/Tiako, particularly his point about Rome being made to fit any ideology necessary. I'd take it a step farther, if I can get away with a generalization, by saying that while the societies that evolved into the Western world continued to point to Rome as an antecedent and use it for their own ideological purposes, Roman cultural legacy was superseded in the East by Islamic identity. This isn't to say that links to Rome didn't still exist in the old eastern provinces after the Muslim conquest, but the Muslim Arabs didn't actually claim descent from Rome in any real sense.
I'll add this: I don't think it's accurate that the western provinces were a "periphery" compared to the east. The east was richer and more developed, certainly, but Rome itself was in the west, the empire arose in the west, the Italian senatorial class filled the ranks of the bureaucracy and was thus heavily involved in the administration of the state, roughly half the legions were deployed in the western provinces, and so on. In fairness, Rome is a big topic, and it's hard to get more specific about the east-west balance without knowing which period you're talking about. But in no period of Roman history, with the possible exception of the last few years before the fall of the Western Empire, was the west merely a periphery, firmly secondary to the richer east.
This is not going to answer your interesting question, because I'm curious about this myself...
not only was the Western half of the Empire kind of a periphery to the more developed and urbanized East (Anatolia, Levant, Egypt among other regions)
Do you have more readings on this?