I mis-typed, there have to my knowledge not been any Popes to offer the Eastern Rites in the Present building, there have however been Greek prelates and Patriarchs who have celebrated the Eastern Rites in the presence of the Holy Father at an altar set up near the Papal Altar in the present building. I’ll have to research when the last Greek to be elevated to the Holy See was. I will get back to you on that one.
I have somewhere a book that discusses Popes who have celebrated Eastern and western rites. While most have offered primarily the western or Latin Rite when ascending the throne of Peter, many have from time to time offered the Eastern Rites in St. Peters. Greeks elected to the See of Peter since the first couple of centuries…
Pope Theodore I
Pope from 642 to 649; the date of his birth is unknown. He was a Greek of Jerusalem and the son of a bishop, Theodore. His election as pope was promptly confirmed by the Exarch of Ravenna, perhaps because he was a Greek, and he was consecrated 24 Nov., 642.
Pope St. Martin I
Martyr, born at Todi on the Tiber, son of Fabricius; elected Pope at Rome, 21 July, 649, Martin, one of the noblest figures in a long line of Roman pontiffs (Hodgkin, “Italy”, VI, 268) was, according to his biographer Theodore (Mai, “Spicil. Rom.”, IV 293) of noble birth, a great student, of commanding intelligence, of profound learning, and of great charity to the poor. Piazza, II 45 7 states that he belonged to the order of St. Basil. He governed the Church at a time when the leaders of the Monothelite heresy, supported by the emperor, were making most strenuous efforts to spread their tenets in the East and West.
Pope John VI
A Greek, the date of whose birth is unknown; d. 11 January, 705. He ascended the papal throne 30 October, 701.
Consecrated 25 March, 708; d. 9 April, 715; a Syrian, the son of John, and “a remarkably affable man”. The first half of his reign was marked by a cruel famine in Rome, the second by an extraordinary abundance.
Pope St. Zachary
Reigned 741-52. Year of birth unknown; died in March, 752. Zachary sprang from a Greek family living in Calabria; his father, according to the “Liber Pontificalis”, was called Polichronius.