The following is from Norval Geldenhuys' commentary The Gospel of Luke in the old New International Commentary on the New Testament series. Geldenhuys was a Dutch Reformed minister in South Africa. His work is quite dated (1951), and my understanding is Joel B. Green's replacement is better, but still I think there's some useful information on the topic, though there may be disagreements in interpretation:
Fortunately, from very early times in the history of the Christian Church, there exists straightforward evidence that Luke2 was Paul's fellow-traveller who wrote the Gospel and Acts3. The anti-Marcionite Prologue to the Third Gospel (between A.D. 160 and 180), which survives in both Greek and Latin, gives the following account:"Luke was an Antiochian of Syria, a physician by profession. He was a disciple of the apostles and later accompanied Paul until the latter's martyrdom. He served the Lord without distraction [or 'without blame'], having neither wife nor children, and at the age of eighty-four he fell asleep in Boeotia, full of the Holy Spirit. While there were already Gospels previously in existence - that according to Matthew written in Judaea and that according to Mark in Italy - Luke, moved by the Holy Spirit, composed the whole of this Gospel in the parts about Achaia. In his prologue he makes this very point clear, that other Gospels had been written before his, and that it was necessary to expound to the Gentile believers the accurate account of the [divine] dispensation, so that they should not be perverted by Jewish fables, nor be deceived by heretical and vain imaginations and thus err from the truth. And so right at the beginning he relates for us the nativity of John - a most essential matter, for John is the beginning of the Gospel, being our Lord's forerunner and companion both in the preparation of the Gospel and in the administration of the baptism and the fellowship of the Spirit. This ministry [of John] had been mentioned by one of the Twelve Prophets [i.e. Malachi]. And afterwards the same Luke wrote the Acts of the Apostles."
The statement by numerous church fathers and by Paul that Luke was a physician is also corroborated in the Gospel and in Acts. In 1882 W. K. Hobart, in The Medical Language of St. Luke, defended the thesis that the third Gospel and Acts are permeated by the medical terminology current during the first century. Harnack, Zahn, and Moffatt also, after a careful sifting of Hobart's data, came to the conclusion that the author of Luke and Acts was a physician. Later on Cadbury4, who is exceptionally critical and unwilling to assume Luke to have been the author of the Gospel and of Acts, maintained that the so-called medical words and terms in Luke and Acts also occur in the non-medical writers like Lucian and Josephus and that in those days there existed no noticeable difference between the technical and non-technical language.
Cadbury is right to the extent that the language of Luke and Acts does not of itself prove that the author was a physician. Nevertheless the fact remains that the language and terminology of Luke and Acts are of such a nature that they corroborate5 in a striking manner the tradition that the author was Luke the physician. The following may be cited as a number of examples of medically tinted language and terminology from Luke: Luke iv. 38 describes the disease of Peter's mother-in-law as a "great fever", while Mark merely describes it as "fever". Now it is a well-known fact that medical writers of those times were accustomed to describe fever as a "small" or as a "great" fever.6
Luke v. 12 describes the leper as "a man full of leprosy", while Mark and Matthew merely say "a leprous man". Here also the expression of Luke is typically medical, as is evident from the writings of Hippocrates.7
In the same way the precise manner in which Luke describes different cases of disease (e.g. xiii. 11, viii. 42, vi. 18, xiii. 32; Acts vi. 22, ix. 33, etc.) fits in with the fact that he was a physician.8
Taking all the data into consideration, one cannot but come to the conclusion that, although the language and style do not per se prove that the author of the books was a physician, the statement by Paul9 in Colossians iv. 14, and the unanimous assertions of the ancient church fathers that Luke was a physician, are clearly corroborated by the nature of the contents of the books.10
2 That the name "Lukas" is the abbreviated form of "Lukios" is generally accepted (cf. Deissmann, Light from the Ancient East, p. 435). Zahn, however, thinks it is the abbreviation of Lucanus (Introduction to the New Testament, iii, p. 5).
3 Cf. Jülicher-Fascher, An Introduction to the New Testament, p. 330.
4 The Style and Literary Method of Luke, pp. 39-72.
5 Plummer, Moffatt, Creed and several others also favour this view.
6 Cf. Creed, op. cit., p. xx.
8 It seems probable that Luke, even after he became a companion of Paul, continued with his practice as a physician. "It is also possible that he rendered valuable services as a physician to the apostle himself, who was often severely ill" (Zahn, Introduction to the New Testament, English transl., vol. iii, p. 1). This would make it clear why Paul calls Luke "the beloved physician" (Col. iv. 14). For would he have called him so if he had discarded his practice as a physician years before? And by calling him the beloved physician, does that not point to personal gratitude Paul felt towards him for services rendered to him by his physician companion?
9 That the Luke of Colossians, iv. 14 is the same as the author of Acts (and so of the Gospel) "is completely established by the content [of Acts], the thoroughly Pauline conception of Christianity, the accurate acquaintance with Paul's fortunes and the central role which is accorded to Paul" (Ed. Meyer, Ursprung und Anfränge des Christentums, i, p. 3). Only a companion of Paul, and a very close companion at that, could and would have written the book of Acts, in which so much prominence is given to the apostle.
10 Cf., for a fuller exposition, Harnack, Luke the Physician, pp. 175-98; and Ramsay, Luke the Physician, p. 16.