The short answer is that Rome defeated all other potential city state/empires during its rise to power. There was no guiding intelligence within Rome that set out to build an Empire, rather at each point in time the Consuls then Emperors took decisions that involved the defeat of potential enemies and the annexation of their territory, which opened Rome up to further potential threats until they were left managing a huge empire. Please excuse the very brief sketch.
If Alexander had turned his attentions West not East, Rome may well have been a footnote. As it was the Macedonian successors to Alexander were unable to preserve his empire intact and it split into ultimately conquerable smaller empires (Macedonia and Europe, the Seleucids in Asia and the Ptolemies in Egypt). The (very) short explanation for this is that the Macedonian leaders had personal followings, binding the men on whom they relied to themselves (Errington - A History of the Hellenistic World p68) rather than to their city, as in the early/middle Roman Republic.
Rome was spared having to deal with the Macedonians at a time (c340-290 BC) when they were able to consolidate their dominance in the Italian peninsula (Baker -Ancient Rome p41) and emerge as the leading city state in Italy.
Next came Carthage (260-146BC). In the three Punic Wars Rome was able to first drive the Carthaginians from Sicily and then Spain and North Africa. This was no forgone conclusion and was an existential struggle in which Rome suffered serious defeats (such as Cannae 216 BC) and was nearly destroyed itself. The Carthaginians were preeminent at sea with a huge unrivalled navy in the Mediterranean. People think of the Romans as invincible in battle. It wasn't the case. They lost huge number of men in land and sea battles -massive defeats occurred. What set them apart was a willingness to throw more and more men into the fight, replacing defeated armies and navies over and over (Richard Miles - Carthage Must be Destroyed). A short explanation for this is that in the early to middle Republic Consuls only held power for a year and were acutely aware that they only had a brief opportunity to make a name for themselves. War could be hugely profitable following the sale of captured slaves and the opening of foreign markets. The pool of manpower in the peninsula and the political advantages of service in the Roman army gave them the raw materials to keep trying.
The war with Carthage segued into a war against the Greeks and Macedonians. The Macedonians had backed the Carthaginians and so were punished at the battle of Pydna 168 BC and Rome moved into Greece to ensure the resulting power vacuum was to their liking - defeating the Corinthians at the same time as the Carthaginians in 146 BC.
The now free cities of Western Asia Minor were swallowed. Here it was Rome's businessmen, the equestrian class who led the charge, looking to exploit markets with Rome's military and legal backing. Then the Seleucids' dynasty effectively collapsed with the death of Antiochos Sidetes in 129 BC. Syria and the Eastern Mediterranean was open. Julius Caesar took Gaul by 53BC. That only left the Ptolemies and Cleopatra signed Egypt's death warrant by backing Mark Anthony over Octavian, culminating in the battle of Actium 31 BC and Augustus' annexation of Egypt as a personal province of the Emperor.
Rome was now an Empire and attention turned from the Mediterranean to the Rhine and Danube frontiers. The idea was to secure Rome's safety rather than embark on expansion for its own sake. Successive Emperors developed Augustus' international strategy exactly as he did his domestic - without a grand master plan. It was all improvisation done to secure Rome's borders.
I think there's two separate questions here: a) why was Rome able to unite the Mediterranean (the Mediterranean being a better unit than artificial ideas like "Europe" and "Near East) and b) why did no other city state do so?
A) is a question whose answer would depend on specific features of Roman political/military organization and economy as well as the specifics of the economic, political, and social situation of the Mediterranean in the few centuries before Yēšūa^(c). This is what previous people have tried to answer.
However, b) is what I'd like to address and it requires first a bit of reframing and then some thinking a bit about broad-strokes, long-term history (which in turn brings up questions about how deterministic history is). I'm much more receptive to this kind of thinking than many, and I rather like the work of Ian Morris and, god forbid, Turchin (please don't chase me out with torches and pitchforks).
To start with the reframing, asking why Rome, and no one else, united the Mediterranean is a bit like asking why you always find what you're looking for at the last place you look. Once you find what you're looking for, you don't keep looking. Similarly, once a first millennium BC Mediterranean city state conquers the rim and turns it into a cohesive empire, no other first millennium BC Mediterranean city state can do so.
That's the first part. The next part is asking why the Mediterranean should be united at all. Very, very, very roughly speaking, the sea and shared subtropical climate made transport and communication relatively easy (by ancient world standards) and two and a half thousand years ago it was possible for a single state to muster up the resources and manpower to conquer it. This isn't to say that history is deterministic; just that there's a few patterns and a bit of cause-and-effect (or cause and some-finite-set-of-possible-effects-of-that-cause) going on. Looking at 600 BC you could say that there's a high-ish probability that the Mediterranean would, in the coming centuries, be unified into a single empire, but you wouldn't be able to get more specific than that. Would the unifiers be Greek or Phoenician, since Greek and Phoenician city states were so numerous and such important powers at the time? Or would it be someone else entirely? Once unified, what trajectory would that unified empire take?
Once you get to that point you can start to look at the specifics of Roman politics, economics, and military that others have brought up.
I hope, if nothing else, I've helped you think a bit about what you're asking with your question!
Well, initially the Romans weren't particularly special they were just dogged. They did adopt the Greek style of fighting and then alter it to make an incredibly effective fighting force (and later alterations were also key as well as the transition to a standing army). But it took them quite a long time to conquer most of Italy. When the Romans were first conquering Italy or then their war with Carthage they would be defeated time and again. But they just wouldn't give up. For example, when their navy was destroyed during the first Carthaginian war they just built a new one. They could lose every battle they faced in the second war but they'd just regroup. In this way, they slowly but surely conquered the surrounding lands.
By contrast, a place like Carthage was hiring expensive mercenaries and couldn't really afford to pick themselves up every time they fell.
As success began to build up then they started to get an impression of their own special role and destiny. But it certainly didn't start that way.
Another aspect of Rome was that they had a good balance of traditionalism and willingness to adapt. An example of this is famously how they took much of Greek culture and made it Roman. But they did this constantly. Rome might conquer another city-state that already had a legislative body and just turn that body with largely the same members into a Roman like senate to handle local affairs and answerable to the Roman governor of the region. They may adopt local traditions or local gods. The point being that Rome wasn't necessarily coming in and just saying "this is Rome now". For your average person living in Antioch or Athens there were new roads, baths, temples, and so forth. And there were Roman legions around. Plus, the new taxes. But your everyday life didn't change much. The elites in your area didn't change much. There wasn't really much of a reason to rebel. For an early example, look at the Latin wars. These weren't wars where the rest of Italy wanted to leave Roman control. These were just wars where they wanted the benefits of Roman citizenship. That's very telling of how it played out elsewhere. Another telling example is how many Roman emperors after the fall of Augustus's dynasty came from the provinces rather than Rome or even Italy. Those areas had been firmly absorbed into the empire. The ability to do that was key to long term success.
Now that is more true in already "civilized" places that Rome conquered. Like North Africa and Greece. It was less true in Hispania, Gaul, or Britain, initially. And in those places rebellions were more likely. But even then once pacified they tended to get with the program. When Britain (which did have large scale rebellions) was finally abandoned they wrote and begged the Emperor to send help. Part of Rome's success was just simply that people usually didn't want to leave Rome once they were conquered. Romans were brutal conquerors but relatively benevolent lords. They provided roads, aqueducts, a huge economy for trade, troops for protection. Don't get me wrong there were downsides. Before Augustus governors would bleed the provinces dry. And even after that was a concern. There were taxes. And then after Diocletian the local obligations became even more onerous. But, overall, being a Roman subject wasn't such a bad deal. Rome was successful because many people liked being under Rome. Really Rome could have fallen in the third century and it didn't for many reasons. But a big part of it was people liked being connected to Rome. Provinces weren't necessarily anxious to leave, with exceptions. Rome wasn't constantly putting down internal rebellions while also fighting the Germans or the Persians or whomever and that mattered a lot. (Although I can't stress enough that I'm not saying rebellions never happened. They did. Just relatively less than you'd think for such an extensive and long-lived empire. And there were many civil wars with Roman legions fighting Roman legions. But that is different from a locally based uprising of native people.)
I'd also say Rome had effective organization and administration. They made things as efficient as possible. They had roads, they had post stops for messengers, they built bridges. So they could move troops relatively quickly. And they were made aware of threats relatively quickly. And their social structure, which was based around the entire concept of patronage, made sure that they had a ready pool of officials who felt obligation to serve the state (this wasn't unique in city-states though). Obligation and service was built into the Roman society. And they created a for advancement. So, in the first and second centuries a Roman emperor could be a disaster and it would have little to no impact on the provinces. Or in the late Republic when there were civil wars for decades the provinces were still being run. Because their local government could run fairly effectively without constant oversight. (By the late second and third century a hands on emperor became more crucial because of outside pressures and the increased need for tax income.) After Diocletian the bureaucracy changed a great deal but still effective administration was key.
Long post short, Rome had an effective military and, at least in the early periods, were always willing to rebuild and come back fighting when they were faced with defeat. Their military success helped them conquer the surrounding lands. Their willingness to somewhat accommodate local culture and elites (at least in the "civilized" areas), their tendency to build wherever they conquered, and their relatively effective rule made many (not all) of the places they conquered satisfied with their rule and unwilling to rebel. Or unwilling to rebel more than once or twice. As time went on these two prongs fed each other. Their increased land provided more troops, greater resources, and safe places to attack from. Most of that wasn't unique to Rome but few places had been able to use it to their full advantage in the way Rome did. By the end, when Rome was disintegrating in the west, the idea of Rome itself had become very powerful and people clung to it.