This is a very interesting question, because it shows how the world is all interconnected. Could the Romans develop steam engines? Let's look first at the technological requirements and their implications.
The Romans knew perfectly well that there's force in steam. There were even some crude demonstrative devices, such as the aeolipile mentioned in the question. So what more was needed, from a purely technological point of view?
First, they needed fuel. The classical Greco-Roman civilization used relatively little fuel, and that fuel was wood and charcoal. They used little fueld because they lived around the Mediterranean, where winters are mild, and they lived during a warm period. The Mediterranean region has few extensive forests, and it has essentially no mineral coal.
Since there is no coal around the Mediterranean, the Romans had a very vague idea of what coal was, or how to use it. They knew that in Britain one could find some black rocks which burned, but that's about all they knew about coal. But let's say that after somebody building the first engines, some Roman engineer returning from Britain would have experimented with coal, and the Romans would have pushed northwards to tap the extensive coal reserves in Britain and Gaul.
So fuel is not such a big deal; it's perfectly possible to imagine Romans mining coal in Britain.
Second, and here we come to the really difficult technological problems, they would need metal and metalworking techniques to make the engine. This is a big problem. The only suitable metal available in those times was bronze; bronze is definitely not cheap. Why not cheap iron? Well, the problem is that in Roman times they knew how to cast bronze, but nobody had ever seen molten iron, much less attempted to cast it. Bronze remained the only metal suitable for casting for a thousand years after the fall of the Western empire; that's why cannon were made of bronze up to the dawn of the Modern age.
Remember that in the classical world metal was used on a much smaller scale than in the Renaissance and the Modern Age. Metal was precious.
And metalworking techniques were also problematic. The Romans did not have lathes able to machine metal; all they had was hand-powered lathes suitable for turning wood. The first lathes able to machine metal appeared (again) one thousand years after the fall of the Western empire. No lathes means no (reproducible) screws and no way to make accurate cylinders or pistons. Making a steam engine without accurate cylinders and pistons is non-trivial; yes, a crafty artisan could have made one cylinder and one piston, at a great cost, but this does not a navy make.
But the killer problem is metal pipes. They just did not have the technology to make high(-ish) pressure metal pipes. The only metal pipes which they knew how to make were very low pressure lead pipes; don't try to pipe live steam through one of those. Why are pipes important? Without high(-ish) pressure pipes one can make only an atmospheric engine, like Newcomen's original (1712). This kind of engine is suitable for pumping water out of mine shafts and little else.
Implications of the technological requirements
One can of course imagine the Romans solving the technological problems. (Or more likely the Greeks, but then by the 2nd century the Greeks were fully integrated in the empire, which they no longer perceived as an alien domination but rather as the natural culmination of the history of Greece.) The question is then, what kind of world would it have been? Would it have preserved the essence of the classical world?
I would say no. The kind of technological advancements we are speaking of imply a world with a highly qualified workforce, with an intense market economy, with systematic technological training, with widespread use of metal parts, with precise measurements... This is not the classical world. It just isn't.
Could they have done it while preserving something of the classical world? Yes, of course! After all, we did it -- we still have senators, who (in the U.S.A.) meet on the Capitol hill, we still have tribunals, and (in many European countries) we still have prefects and quaestors, and our languages are full of Latin words. If you are content with a world which preserves some (maybe most) of the forms of the Roman world but little of its essence, then yes, this is imaginable.
So why didn't they?
But the Romans did not develop the technologies necessary to make steam engines. They did not develop the kind of intense market economy where steam engines would have been useful. Why? Because they couldn't. The Romans did invent the state in the modern conception of the word, they did invent the concept of uniform laws, and codes, and bureaucracy, and public service, and public infrastructure and many other aspects of civilization which in the fullness of time made the modern world possible. But one thing they did not invent -- a functional and stable mechanism for the transfer of power.
Peaceful transfer of power was always a goal: most emperors were decent people, who really strived to preserve and develop the empire. Most of them actually succeeded in transferring power to their successor peacefully; for example the chain Nerva–Trajan–Hadrian–Antoninus Pius–Marcus Aurelius (the adoptive Nerva-Antonine dynasty, the "five good emperors") provided stabilty and good governance for an 84 years, almost a century, for 96 to 180 CE. But once in a while the transfer of power failed, or the successor was poorly chosen, and havoc ensued.
So the empire had periods of peace and development -- under the Julio-Claudians, under the Flavians, under the Nerva–Antonine dynasty -- and periods of anarchy and civil strife. And in the 3rd century one of those periods of anarchy -- the Crisis of the Third Century -- tilted the balance too much to the side of darkness. The empire was fatally wounded; it would never recover.
It could have been avoided, especially in fiction. In fiction, one can always make Commodus a good emperor; one could imagine a rule establishing that imperial succession must go to an adopted heir; or at least one could imagine the Romans instituting legallythe office of emperor and making it hereditary. (One of the big issues with the Roman empire was that before Diocletian it was not legally and empire and did not legally have an emperor; the state was officially a republic, and the men whom we call emperors ruled by cumulating key positions in the state; it's complicated, but basically they cumulated the position of speaker of the Senate, commander in chief of the army, high priest, and most importantly tribune of the people.)
You must make the economy of the Roman empire much more intense; steam engines are no use in a sleepy economy.
You must somehow overcome the Greco-Roman disdain for artisans; maybe have Alexandria develop into a hotbed of industry, with Rome forced to compete, I don't know.
You must avoid the crisis of the 3rd century at all costs, even if that means instituting the dominate early.
You must imagine a chain of unlikely but not impossible inventions...
And as a result you will get a world which resembles the classical Greco-Roman world a bit, but it's quite different, and you will have changed everything to follow. There will be no Middle Ages, no Islam, no British Empire...