Was the Roman legal system successful in uniting and stabilizing people

I think, maybe. Because,

Long before the Roman Republic was established in 509 BCE, the early Romans lived by laws developed through centuries of custom. This customary law (ius, in Latin) was handed down through generations and was considered by the Romans to be an inherited aspect of their society as it had evolved from its earliest days. Integral to the notion that this customary law was part of the fabric of early Roman culture was the fact that this law only applied to Roman citizens and was thus ius civile, or civil law.

The citizens of Rome were divided into two classes: patricians, the elite class who ruled Roman society, and plebeians, the common people. One element of the patricians' elite status was that a group of patrician men called pontiffs were the ones who made decisions and ruled in questions of customary law. Over time, plebeians came to see that because of the disparity between their positions, patricians tended to have some advantage in the legal decisions made by the pontiffs who were their equals in status and power, and dissatisfaction grew with what many perceived to be the arbitrariness of the decisions made.