As it has been discussed hereabouts recently, theoretically, any baptized male Catholic can be elected by the conclave Cardinals.
However, it has also been pointed out that that person has to be at least a priest, because of the necessity of ordaining him immediately a bishop upon his acceptance of his election as Pope.
Further ahead, it was noted that, realistically, the next Pope will come from the ranks of the College of Cardinals, more specifically, from among the Cardinal electors themselves, 117 of them to be exact. (Cardinal Sin and Cardinal Rivera, retired Archbishops of Manila and Monterrey, respectively, will be unable to attend the Conclave due to failing health leaving the actual number of electors at 115 as of today.)
Among the 115 electors are only 3 Eastern Rite Cardinals: Cardinal Daoud, retired Patriarch of the Syrians; Cardinal Husar, Major Archbishop of the Ukrainians; and Cardinal Vithayathil, Major Archbishop of the Syro-Malabars. Cardinal Sfeir, Patriarch of the Maronites, and Cardinal Ghattas, Patriarch of the Catholic Copts, are both over 80 and, therefore, cannot vote although, theoretically, either is eligible for election.
Numerically, there are 112 Latin Rite Cardinals and only 3 Eastern Rite Cardinals in conclave. Unless there is a very cogent reason for the Cardinals to elect an Easterner, history and tradition show that the next Pope would be from the Latin Rite.
If the Cardinal electors are swayed by the Holy Spirit to elect an Eastern Rite Cardinal, the votes should go to Cardinal Daoud for the following reasons:
(1) he is the most senior among the 3 Easterners, being a member of the Order of Bishops within the College of Cardinals;
(2) he is the ex-Prefect of the Congregation for the Oriental Churches of the Roman Curia and is, therefore, known by most, if not all, of his peers in the College; and
(3) he has been a Patriarch of his particular Church, which shows that he has been a leader in his own right and Rite.
Both the Codes of Canon for the West and the East, define and identify only 1 office from which the supreme authority in the Church is exercised: the Office of the Roman Pontiff. The Latin Code (the Eastern Code contains a similar provision) provide that:
Can. 331 The office (of Roman Pontiff) uniquely committed by the Lord to Peter, the first of the Apostles, and to be transmitted to his successors, abides in the Bishop of the Church of Rome. He is the head of the College of Bishops, the Vicar of Christ, and the Pastor of the universal Church here on earth. Consequently, by virtue of his office, he has supreme, full, immediate and universal ordinary power in the Church, and he can always freely exercise this power.
The “Office of the Roman Pontiff” is what is known as the “Petrine Office” through which the “Petrine ministry” is to be exercised for the benefit of the universal Church here on earth.
Of all the titles emanating from the Pope being the “Roman Pontiff,” conspicuously absent is the title “Patriarch of the West.” Consequently, such title, or office, has been done away with by the Catholic Church or that the Catholic Church has developed into a papal system of governance as opposed to the patriarchal system still extant in the East.
This being the case, there is no problem for an Eastern Rite Cardinal to assume the “throne of Peter” as the Roman Pontiff, or, in today’s parlance, Supreme Pontiff of the universal Church. Yes, he is to be elected as “Bishop of Rome” simply because the “Office of Roman Pontiff” resides in him and from which the title “Pontifex Maximus” or Supreme Pontiff of the universal Church emanates. He need not abandon his own Rite in the process.
As to the effect of an Eastern Pope on our ecumenism efforts with the East, especially the Eastern Orthodox, I refrain from any conjectures for it could be a good subject of a future discussion.