No, Russia is an excellent counter-argument.

Why was the ruler of Russia called Caesar? Because a some culturally Roman guy conquered them, as in JQuinton's narrative? No. Rather, because they converted to Christianity, and so they greatly respected the (Eastern) Roman Empire and saw it as part of their world, and so their rulers started calling themselves Caesar to invoke that heritage. In other words, they took to the Roman (Byzantine) cultural heritage because they became Christian, they did not become Christian because they had Roman or Byzantine heritage.