Note well: a Catholic social system would not necessarily have to be a monarchy. Might be helpful, but not absolutely needed.
Monarchy, or at least hereditary elites, would probably work best. An electorate informed by Catholic principles, and recognizing Christ as King, could be counted upon to use democracy to apply these principles and to elect leaders who would rule wisely and with goodness and holiness. I would hate to see dictatorships arise.
Are you serious!!! Just look up in Iran what is the status of political opponents (hint… google Evin Prison), some of the religious minorities (hint google Baha’i persecution) and people committing common crimes (hint google public hanging and stoning). Nice right?
@Pitcairn17 set me right on this. Iran isn’t a model we’d want to emulate. I was interpreting “mild Iran” as being,
mutatis mutandis, a Catholic social order that would function in accord with papal social principles, with the Church having a prominent teaching role but not directly involved with the government. Many people in Iran may have good, decent, happy lives, but some obviously do not, and are subjected to punishments and atrocities that could never exist in a Catholic social order.