Essentially: No, there's no evidence for that.

The most obvious problem with that statement is that it's a manifestation of a "tech tree" understanding of science and technology, the idea being that every civilization essentially develops in the same kind of way and thus discovers the same technology. Hell, even the terminology is at fault; technologies are "discovered" as if they exist independently of humans in nature, rather than being inventions of actual people who have goals in mind. The idea of steam power as some kind of "goal" that societies work towards is simply a misunderstanding of how history works. It's akin to thinking of humans as an "end goal" of Darwinian evolution.

There's been a huge amount of literature on the development of steam power, and it is true that the idea of using steam to create motion has been around since Greek and Roman times, a notable example being the Aeolipile but that is a toy and cannot do anything useful. Medieval organs also took advantage of compressing and expanding air, and occasionally also used steam. But something that we could consider a "steam engine", i.e. a labor saving device that uses steam, really only turns up towards the end of the 17th century, most notably the one made by Thomas Savery to pump water out of mines. This really is the beginning of what we would call steam power; the use of steam not just as a toy but as a productive device. There is also a large amount of history of economics literature which deals with why steam power caught on at that particular time and place, but I'm not qualified to talk about that so I won't, maybe someone else can. I will say though that the existence of slavery in Rome probably meant that they didn't need a labor saving device the way Early Modern Europe did.

What your friend may be referring to is the idea that the Romans already used all the elements of a steam engine: pistons, crank and shaft, valves, gearing, etc, but had not yet put them together. I guess he supposes that the Romans, if they had more time before being destroyed by the anti-intellectual barbarians who plunged Europe into the Dark Ages (/s), would have put those inventions together into a steam engine. But the medievals had all these things too, and so did several other societies, and they didn't make a steam engine. Unless you can show that the circumstances which led to the development of the steam engine in 17th and 18th century Europe were also present in Ancient Rome, there's no reason to believe that the Romans would have created a steam engine.

EDIT: /u/Hydrogenjoule also points out that there are a variety of technologies necessary to create steam engines, like accurate machine tools, that did not exist until the 18th century.

the use of steam power in the early industrial revolution is overrated. Much early progress was done with water power. Look up Richard Arkwright and the textile mills he setup, all water power. In the US, the Lowell textile mills, the largest manufacturing center in the US, was water powered well into the 19th century. One area i'm interested in reading about is the use of bellows in iron production. It appears in the 16-17th century England water powered bellows were critical in helping establish England's lead in iron production. I think a more productive line of thought is how much did other countries before or during the early industrial revolution use water power that way or in other interesting ways? Also, check out what the Dutch were doing with wind power in the Zaan district in the 14-16th centuries.

EDIT : don't knock water power, its the source of 16% of global electrical production, although its not my favorite energy source, there are environmental impacts.