I don't think the OP was inviting a discussion thread on the relative merits or lack thereof of either a republic or a monarchy. Rather, he seemed to imply that Orthodoxy was linked to a monarchist approach and he inquired as to why that may be so.
Isa said it best in that the Gospel must take a front seat to the debate over secular, temporal political systems.
I would just remind all that my grandparents, and the grandparents of many - if not most of the earliest Orthodox in the United States - fled from the jackboot and fist of oppressive monarchies - the Hapsburghs, the Romanovs or the Ottomans. While it is true that the White Russian emigres had great influence on parts of the Russian Orthodox church here after the revolution (particularly what is now ROCOR), the truth remains that most of the founders of what is now the OCA were former Greek Catholics fleeing the Hungarians. Anyone professing monarchist leanings at an early 20th century so-called Russian Club, Lemko Hall, Carpathian Club, Ukrainian Hall etc...would have been shown the door after receiving a solid thrashing.