It seems the Melkite Church uses the Gregorian calendar.
From the Melkite website:
THE CALENDAR AND THE CHURCHES
Today there are a number of
Churches employing the Julian Calendar including the Armenian Patriarchate of Jerusalem, the Coptic Orthodox Church, the Eastern Orthodox Churches of Jerusalem, Russia, Serbia, Poland, Macedonia, Georgia and Ukraine, and the Ethiopian Orthodox and Catholic Churches as well as the Assyrian and Greek Old Calendarists. Greek Catholics in Carpathia, Slovakia and Ukraine generally follow the Julian Calendar although most of their parishes in the West follow the Gregorian. The
Churches employing the Gregorian Calendar include the Armenian Church, the Church of the East, the Eastern Orthodox Churches of Estonia and Finland, the Malankara Syrian Orthodox Church and most Eastern Catholics (Chaldeans, Maronites, etc.). A third group of
Churches employ the Mixed (“Revised Julian”) Calendar: the Syriac Orthodox Church and the Eastern Orthodox Churches of Constantinople, Alexandria, Antioch, Greece, Cyprus, Romania, Poland and the Orthodox Church in America (although some Polish and OCA parishes are permitted to use either calendar).
The Melkite Greek Catholic Church generally follows the Gregorian Calendar. In countries with an Orthodox majority it follows the Mixed (Revised Julian) Calendar.
