Melkites mean emperors men (or something to that effect). It was used by the non-chalcedonians as a pejorative to describe the Chalcedonian Orthodox since we represented the imperial church of the empire .
The Melkite catholics of today were the Orthodox in the Levant who converted to the papacy officially in 1724 after Jesuit, and Franciscan monks began arriving in the area about a century earlier.
They started sending Orthodox clergy to Rome to be educated for free. After the 1724 union they officially took on the name Melkite to distinguish themselves from the Syriac catholics (non chalcedonian orientals) who joined the papacy a few decades earlier through similar means. In fact the vatican established the missionary group the propaganda for the promotion of the faith specifically to target Syrian and Lebanese christians:
Congregation for the Evangelization of Peoples - Wikipedia
Today this new Ukranian structure are kindred spirits to the "greek catholics" whom are in communion with the melkites.