Why did the Catholic Church divide into the Eastern Orthodox and Roman Catholic Church?

The Catholic Church did not divide, the various "Orthodox" Churches left, came back, left, over a long period of time, mostly due to political reasons. The "Roman" Catholic Church does not exist - "Roman" is just a slur that the English became fond of after the protestant revolt in England. It is only used in English and never by the official Catholic Church


from A Catholic Dictionary, edited by Donald Attwater, Second edition, revised 1957

The Schism of the East

the estrangement and severance from the Holy See of what is now called the Orthodox Eastern Church was a gradual process extending over centuries. After a number of minor schisms the first serious, though short, break was that of Photius; from then on tension between East and West increased, and the schism of Cerularius occurred in 1054. From then on the breach gradually widened and has been definitive since 1472. There was a formal union from the 2nd Council of Lyons in 1274 until 1282, and a more promising one after the Council of Florence from 1439 to 1472. After the capture of Constantinople it was in the Turkish interest to reopen and widen the breach with the powerful Roman church; the patriarchates of Alexandria, Antioch, and Jerusalem were dragged into this policy, Russia and the Slav churches stood out the longest of any: none of these churches, except Constantinople itself in 1472, formally and definitely broke away from the unity of the Church. But in the course of centuries the schism has set and crystallized into a definite separation from the Holy See of many million people with a true priesthood and valid sacraments. The origins, causes and development of the schism are matters of much complication, still not fully unraveled.


Modern Catholic Dictionary by John A. Hardon, S.J. Doubleday & Co., Inc. Garden City, NY 1980

Separation of the Christian Churches of the East from unity with Rome. The schism was centuries in the making and finally became fixed in 1054, when the Patriarch of Constantinople, Michael Cerularisu (died 1059), was excommunicated by the papal legates for opposing the use of leavened bread by the Latin Church and removing the Pope's name from the diptychs or list of persons to be prayed for in the Eucharistic liturgy. A temporary reunion with Rome was effected by the Second Council of Lyons (1274) and the Council of Florence (1439) but never stabilized.