No Ottoman Emperor ever converted to Catholicism. They were all Muslims.
Osman I
The Ottoman Empire provided a limited freedom of religion in that Jews and Christians could live in the Ottoman Empire as second-class citizens (as opposed to being forcibly converted or exiled).
When Constantine converted Christian, he merged the Christian church with the Roman Empire. Thus, Roman Catholicism.
The Byzantine Empire fell in 1453 when Mehmet II el-Fatih, the Ottoman Emperor, conquered Constantinople.
No, he believed it was his duty to protect Catholicism against the Muslims of the Ottoman Empire, the Protestant countries of Europe (England, Scotland, Netherlands, Switzerland), and their allies (France).
There was not a Byzantine king. There was a Byzantine Emperor. The last Byzantine emperor was Constantine XI Palaiologos. He died in battle when Constantinople, the capital of the empire, fell to the Ottoman Turks.
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After the Emperor Constantine converted to Christianity, that became the official religion of the Roman Empire.
The Germanic peoples who invaded the Western Roman Empire were either already Arian Christians or converted to Roman Catholicism. Emperor Charlemagne conquered norther Germany and sent priests to covert the peoples there to Christianity. Missionaries travelled around the pagan parts of Europe to spread Christianity.
The Romans spread two different religions: pagan religion and Catholicism. In the early days of the Roman empire, they spread pagan Roman religion that is usually referred today as Roman Mythology. In 312 AD, Emperor Constantine converted to Catholicism, and made it the state religion in 313. After that point Catholicism was spread by the sword across the Roman Empire. This practice continued until the fall of the Roman Empire, and was picked up by most of the kingdoms that rose out of that collapse. The general way of spreading religion within the Roman Empire, and even more so by the governments that followed the fall of the Roman Empire, was if you did not convert to Catholicism, you died.
Emperor Constantine was a former pagan who converted to Christianity after having a dream of the Chi-Rho (☧now a prominent insignia in Roman Catholicism) and was told to put the "heavenly divine symbol" on the shields of his soldiers, our Blessed Lord telling him, "In this sign, you shall conquer."
The Ottoman Empire was populated by the