Medieval Greece became a small regional power after the restoration to power in 1261, before that it was a big regional power with influence stretching far from its borders in all directions.
Not a Greek state, as the "Catalan" part kinda gives away.
I have a hard time seeing how you could sufficiently build a "Greek" identity separate from a "Roman" one, especially as "Greek" is in this period synonymous with "pagan" and "Roman" with "Christian". You could have that subtly alter over time, so "Greek" comes to be thought of as perhaps meaning something closer to "elite intellectual", but that's hardly a widespread and popular basis for a Greek state.