A strong Egypt or a Pontic empire seem like the candidates to me. The idea is that the Eastern Med is space that needs filling, and by a very strong power that can keep on fighting off Roman attempts to horn in.
It would also be possible to preempt Roman ambitions to the East if a strong empire existed there that simply deterred them from ever trying to move eastward. But people seem to be answering in terms of the latest possible divergence. By the time Mithradates was trying to build the Pontic state for instance the Romans had already stuck a lot of feelers eastward. I've enjoyed quite an interesting TL where the Pontic Empire does occupy the Eastern Med. The key there was for the dynasty to present itself as friend and liberator of Hellenes, while also upholding the dignity of non-Hellenes. But the role of protector of Hellas helps explain the stable frontier on the Adriatic, while a combination of naval power and coastal holdings in North Africa takes the same longitude as division between east and west Med. Bulwarked on the west, the dynasty focused on expansion eastward to take hegemony over Persian sphere of influence, which frankly disappointed me as I was hoping to see the empire sprawl north into the grain lands of southeastern Europe (Ukraine mainly, but going into Europe from that direction, and north to Russia) and south into east Africa. Never mind the extensions though; an empire that simply stuck to OTL Roman eastern boundaries and stopped at the Adriatic to the west would be quite a large and rich empire, and with those bounds Anatolia is a logical center.
The thing is, a purely western Roman Empire would be perceived, and perceive itself, as secondary, peripheral, poor and rustic. No doubt Romans could rework some of this identity into a matter of pride, but culturally they'd be dismissed as semi barbarian outliers, and Roman ambition would always turn toward prospects of seizing richer lands to the east. They'd be probing constantly, any weakness in the eastern state such as a dynastic civil war or ethnic fracturing--or being drained and weakened by barbarian invasions on any front--would lead to Romans attempting to exploit it, penetrating anywhere they can. For the situation to prevail for half a thousand years or longer both empires have to be in good health.
The reason the East is richer is that its resources are better developed--it takes thousands of years to scout out workable mines; agricultural development has been going on intensively for a very long time; a "central" location tends to perpetuate itself as it sits in the middle of its own offshoots. Most of all it supports populations who have developed extensive craft skills over thousands of years. The Romans might import some of the latter and hope that western crafts can develop to match eastern ones, but they are still on the far edge of the great Old World Trade network rather than in the inner part of it; the fact that the potential mines and so forth of the west are largely unknown means that over time the land will become mineralogically richer due to having not been depleted let, but undiscovered deposits might as well not exist until they are found! Forms of agriculture have in fact existed among the native peoples for thousands of years but the population density is low, settlements are scattered, and the practices are not intensive enough to support really large populations. Changing any of that simply requires time, time for local people to innovate new practices while working in versions of inventions from afar they learn of, time for the front line of more or less integrated civilization, united by trade, to spread farther afield--and with the conquest of Gaul and Britain, the Romans would be pretty much stuck, because I don't think it is plausible anyone crosses the Atlantic and develops the ability to return reliably for another thousand years or so, so they are stuck with Europe. The ability to sail down the African coast and then return reliably is a problem of comparable difficulty as reaching the New World, so the useful parts of Africa are also already taken and developed, for what they are worth. Which is quite a lot in Classical times; I gather Roman-run settlements were more predominant and populous in North Africa than in southwest Europe, and unlike the marches to the north they didn't face tremendous barbarian pressure either.
So, a Roman Empire of the western Med, which I suppose would logically claim coastal northwest Africa from whatever Libyan border is established with the power blocking expansion to the East, and extending north much as OTL to limnes along the Rhine, maybe going farther to annex more German lands either on the coast or inland, would be rustic and would rusticate! Nothing but time would lay the groundwork for populations as dense and industrious as those found in the High Middle Ages--and by world civilization standards, Medieval Europe at its best was still provincial and marginal.
Frankly, I don't see how or why the western Empire lasts, lacking the deep pockets that conquest of the east brought the OTL Empire. One might well doubt it would form at all. On a map, it seems evident that Italians could focus on securing the vast swathes of European land to the north, and to modern perceptions this looks like a more sensible move than going south over seas to muck around with Africa and Iberia--but the latter is the sandbox Roman imperialism played in, not Gaul, and long before Julius Caesar subdued Gaul, Roman outposts making the Republic quite rich had been established on the far eastern shores of the Med. If those were preempted and denied, who knows whether the Republic would have the stamina to break Carthage or secure the vast northwestern buffer zone of Gaul and Britain?
Civilization and integration of more and more intensive economies into the broader Old World trade network would presumably go on with or without an empire to formally organize the process; conceivably without a central empire to police things some forms of cultural diffusion might happen all the faster. But unless the eastern empire, whatever that is, takes it into its head to unify control of all the Med into one rule, and surges west to secure the western seas, it may be that the region to the west of the Adriatic never knows political unification at all.