Gregorian calendar was a reform of Julian calendar and was introduced in 1582 by Pope Gregory XIII. Catholic countries adopted it that year. Protestant and Orthodox countries were slow to adopt Gregorian calendar but eventually did. USSR ruled by communists adopted Gregorian calendar in 1927.
Orthodox Churches have persisted with Julian calendar. That creates problems for their followers as they have to follow two calendars. Celebration of Christmas is an example. The governments follow Gregorian calendar and what is 25 December according to Julian calendar is 7 January in Gregorian calendar.
Russian Orthodox Church is the leading Orthodox Church. Among patriarchs the Patriarch of Constantinople, known as Ecumenical Patriarch, is the leader. The survival of the institution of Ecumenical Patriarch is in question and CNN recently carried a programme titled The Last Patriarch.
Constantinople was known as New Rome. In 1054 Constantinople broke away from Rome. In 1439 at the Council of Florence, Catholics and Orthodox unity was achieved. Russian Orthodox Church opposed unity and declared Moscow Third Rome. In 1453 Turks conquered Constantinople and after some years other Orthodox Churches broke away from Rome.
What was possible in 1439 can be possible again. It is good if Orthodox Churches give up their resistance to Gregorian calendar and adopt it. If not, they will be the losers. It will become increasing difficult for their followers to maintain two calendars.