No.
Only Orthodox can be Patriarchs.
Aside from that, if there were union, the Pope is a Patriarch, the precedent is only of Patriarchs of a See being named Patriarchs of greater Sees, never of lesser Sees. Since the Catholic Assertion is that Rome is the Greatest See, and since the EO generally agree with its pre-eminance, even in unity it would be quite out of the ordinary for a Patriarch to take on a lesser title.