No. The Patriarch and the the Eastern Roman Emperor, were always different titles and jobs. Sometimes they might have influenced the other more strongly, but even going into the Ottoman Empire, the Patriarch was largely subservient* to the Emperor (or later Padishah), who chose the Patriarch himself (and with consultation).

What your friend is probably talking about, is the fact that the Patriarchate and the Emperor were seen as two sides of the same coin, which is called: Caesaropapism, where the "secular" ruler, also held power within the Church. However in this case it was the Patriarch who administered to his flock, not the Emperor (unlike the Caliph, who could push out religious interpretations). However it should be noted that the notion of "Caesaropapism", declined over the course of the Empires' power (in my opinion, with its height under Constantine).

*Not the best word, but basically the Patriarch was no Pope, with his own secular power. However there were cases where the Patriarch could definitely influence things.

Not sure how to give an "anti-answer" source. But a simple search on wikipedia will show you all the Patriarchs of Constantinople.

A book I enjoyed on Eastern Byzantine History is Judith Herrin's book: Byzantium: The Surprising Life of a Medieval Empire

As for the Ottoman Empire, Mark Mazower's "A Short History of the Balkans", gives a nice short history about the future of the Patriarchate in the Ottoman Empire.