Melchite, also spelled Melkite, are any of the Christians of Syria and Egypt who accepted the ruling of the Council of Chalcedon (451) affirming the two natures—divine and human—of Christ. Because they shared the theological position of the Byzantine emperor, they were derisively termed Melchites—that is, Royalists or the Emperor’s Men (from Syriac malkā: “king”)—by those who rejected the Chalcedonian definition and believed in only one nature in Christ i.e. the Monophysites.
While the term originally referred only to Egyptian Christians, it came to be used for all Chalcedonians in the Middle East and finally, losing its pejorative tone, came to designate the faithful of the patriarchates of Alexandria, Jerusalem, and especially Antioch.
The Melchite community generally consisted of Greek colonists and the Arabicized populations of Egypt and Syria. They adopted the Byzantine rite and thus followed Michael Cerularius, patriarch of Constantinople, into schism with Rome in 1054. For several centuries afterward, the patriarch of Antioch attempted reunification with Rome, and a small number of Melchite Catholics emerged.
Melkites though are not related to the Lebanese Maronites. These folks first are in Lebanon, not Egypt, are part of the Church of Antioch, an ancient city which was an important center of early Christianity, but the most important part is that they revered St. Maron (d. 410) as their founder & spiritual leader.
Maron was a monk who moved from Antioch to a mountain in Syria to be a missionary and lead a life of asceticism. St. Maron’s disciples moved into present-day Lebanon in the early fifth century and spread the Gospel throughout the region, so Christian Lebanese were then referred to as Maronites.
While Maronites asset that they were always orthodox Christians in union with the Roman see,there is evidence that for centuries they were Monothelites, followers of the heretical doctrine of Sergius, patriarch of Constantinople, who affirmed that there was a divine but no human will in Christ.
This came up because of their misunderstanding that that Jesus acquiesced to His Father’s will at Gesthemane and did as He was told, without the realization that in his humanity, he asked His Father that “this cup be taken from me” (Luke 22:39-44).
—from the Encylcopaedia Britannica and other sources.