Rule-based societies predate the Romans by a lot of time ; Romans weren't even that good at it, considering how often violence was used as a political tool during the republic.
Wouldn't the Code of Hammurabi have preceded that by quite a few centuries?
That's not to say there's not a grain of truth in this perspective. My impression is that Romans were notable for their pioneering work in the field of what could best be described as "civic pride" - the sense that "civilization" represented progress, the idea of the state as a kind of collective project that elevates everyone and that we should all be proud to participate in. They probably weren't the first to think like this - history is long - but they were the last to do so in a largely secular fashion in the West for a long time, and certainly directly influenced the thoughts of the Enlightenment thinkers who eventually inspired the American Revolution.
S.P.Q.R.
The Code of Hammurabi is best understood as royal propaganda designed to portray Hammurabi as a just king. The famous stele starts with a graphic depiction of him receiving the royal rod and ring from Shamash, the sun god who was emblematic of truth and justice. The laws are best understood as a statement of the kind of justice Hammurabi wished to see done in his kingdom, not a set of rules.
Rule-based society is something different: when the Law itself has a power of it's own, stronger than power of any individual or group. It was first invented in Rome and provided them tremendous advantage, until started to crumble during the Long 3rd Century Crisis.