As many should well know, the worldwide Orthodox Communion officially uses the term Catholic (big "C") in its name in its official documents.
I have been asked about the usage of "Roman Catholic" with regard to all those in communion with Rome. Some claim that "Roman Catholic" refers to rite. However, any and all adherents to the Vatican, whether eastern rite or whatever, are bound to acknowledge the official pronouncements of Vatican I, which state explicitely that "Roman Catholic" refers, not to rite, but to adherence and obedience to the occupier of the See of Rome.
The name "Holy Roman Church" referring to the entire communion in adherence to the Bishop of Rome is found, for example, in the Council of Trent Session 3 "Decree touching the Symbol of Faith," and in Vatican I, Session 2 : 6 January 1870, Profession of Faith .
Later in session 2 of Vatican I, those in communion with the Bishop of Rome of any rite must "acknowledge the holy, catholic, apostolic and Roman church, the mother and mistress of all the churches." Therefore, we also have the following as officially a longer form of its name: "The holy, catholic, apostolic and Roman Church" (Vatican I session 3 Dogmatic Constitution of the Catholic Faith). We find the same phrase all in capitals this time in Mystici Corporis Christi (1943) by Pius XII. However, "Roman church" by itself without anything added is utilized to refer to the local Church in Rome throughout the rest of the council's pronouncements.
We also see in Vatican I it is clearly the "Holy Roman Catholic Church" In fact, we see in the acts of the first Vatican Council that the proposal by a few English speaking Bishops to change its official naming of the Church from "Sancta Romana Catholica Ecclesia" (The Holy Roman Catholic Church) to "Sancta Catholica Ecclesia" (The Holy Catholic Church). The move to eliminate the word “Roman” from the official name was defeated by an overwhelming majority, as was a second vote to even put a comma in between Roman and Catholic (cf. Theodore Granderath. Constitutiones Dogmaticae Sancrosancti Oecumenici Concilii Vaticani. (Herder 1892), pp. 5, 27). The council stuck to the phrase “Roman Catholic Church” for all rites and actually require the acknowledgement of the whole church as “Roman” is session II profession of faith and also in session III dogmatic constitution of the catholic faith.
"Roman Catholic Church" is also the official name of the Church of all rites used by Pius XI in Divini Illius Magistri (1929). In Humani Generis (1950) we likewise find: "the Mystical Body of Christ and the Roman Catholic Church are one and the same thing."
The phrase “Roman Catholic Church” is what is used in the agreements signed by TH; Paul VI and Athenagoras; JP II and Rowan Williams (Anglican), and between Benedict and Bartholomew. It is used in many other places as well. John Paul II himself jointly named it the Mixed Commission for Theological Dialogue between the Roman Catholic Church and the Orthodox Church . This was again out of respect for the Orthodox usage of the name as well.