It is interesting that in the earliest extra-biblical reference to Luke, his status as Jew or Gentile is not given:
Little is known of Luke’s background. The best records suggest he was a native of Antioch, Syria, a Gentile, a physician, a writer, and a devoted Christian. The Anti-Marcionite Prologue to Luke, dating from the second century, states:
Luke was an Antiochian of Syria, a physician by profession. He was a disciple of the apostles and later accompanied Paul until his martyrdom. He served the Lord without distraction, having neither wife nor children, and at the age of eighty-four he fell asleep in Boetia, full of the Holy Spirit.
Richard R. Melick, vol. 32, Philippians, Colissians, Philemon, electronic ed., Logos Library System; The New American Commentary (Nashville: Broadman & Holman Publishers, 2001), 330.