I believe it would have been technically feasible that gun powder could have been developed at least during the Roman Empire era but how exactly would the Romans have used it?
Well, they hadn't the metallurgy to make useful and safe guns and artillery. But they could have made early hand grenades; they could have made explosive onager projectiles; they could have built rudimentary land mines (didn't the Romans love traps and other inventious siege fortifications?); they could have used petards to blow up walls and ships; they could have designed wooden rockets and ballistae propelled by gunpowder like
Hwachas; finally, they could have built fireships and combined Greek Fire with gunpowder.
Also, don't forget the economic impacts of black powder: more effective mining and the possibilty to plow up rock to build tunnels (roads through the Alps) and dig canals (Isthmus of Corinth and Suez canal). So in fact, civil economy might benefit from a military technology. The new civilian infrastructure (roads and canals) would in turn help the army to quickly relocate troops through the empire.
Apparently church bells had something to do with it -- the techniques needed to cast a big bell for a church tower are apparently much the same as those needed to cast a cannon. Non-Christian countries didn't have as much need for big bells, so their metallurgy/the number of people with knowledge of metallurgy was therefore lower.
Finally, the use of gunpowder and the need of advanced metallurgy for military purposes would have forced the Romans to develop better metallurgic techniques, so maybe technollogy might have developed faster if the Romans had had gunpowder.
Did they have the right infrastructure and technological culture to take full advantage?
Well, no:
1) Right infrastructure. The Roman Empire lacked a tradition of chemistry (often religiously shaped by alchemy) - China had one, since chemistry was a way to achieve the Taoist dream of eternal life. But there is still a little chance that a Roman scientist/philosopher accidentally discovers the right formula - but this probability is very low, which is why gunpowder was discovered in China and not in Rome or western Europe.
If you search for a possible person discovering gunpowder in ancient Europe, take
this one.
2) Culture. Roman technological culture wasn't particularly good (heavy wheel plow and water mills, though known in the Roman Empire, didn't spread very quickly through the empire), but the Roman military was open for innovations. Look on how they adopted foreign weapons; now imagine that Zosimos discovers gunpowder during late antiquity - the Roman army would gladly have accepted this new weaponry.
Would gunpower save the Roman Empire? If it's discovered in Byzantine times, it might help the Roman Empire against the Arabs (just like Greek Fire). If it's discovered during the republic or the early empire, it would be only of limited use, and maybe the Romans would even forget it because they don't need it.
However, if the Roman Empire acquired gunpowder during late antiquity, it could be of great use. The Persian cities would be defenseless against Roman gunpowder technology, at least until the Persians adopt gunpowder and use it themselves when besieging Roman cities. The Barbarians would be scared at first and this might be enough to win some decisive battles.
I hope I could help you.