Actually, I don't think it would look like the Eastern Roman Empire. For one thing, I very much doubt that 10% or more of the urban population would be made up of monks. Second, we have obviously shifted quite a bit towards commerce, and away from agriculture. And third, with the way communications and other technologies have advanced, it'd be harder to have all that intrique and murdering leaders and whatnot as acceptable phenomena; people would just leave such an unstable society today, whereas they didn't have that choice a thousand years ago.
I think an Orthodox country, if made up of today's Orthodox, would probably be a Republic, with perhaps a Monarch of some sort who had mostly cultural powers. The worship would be exclusivistic, and ornate. Other religions and belief systems would be tolerated, but hardly have the type of freedom that a government like the U.S. allows. I think that, like the U.S., the culture would be somewhat moderate to conservative, when compared to peer nations, but of course her more conservative/traditional enemies would still find lots of reasons to cast stones. I think political contests, sporting events, etc. would be hotly (and sometimes violently) debated and contested, reflecting the ethnic and cultural divisions among the population at large. Probably less than half the population would attend Church on a regular bases, possibly significantly less. Still, the necessity of religion, going to monasteries, etc. would consistently be rated as being very important in polls.