According to the Catholic church, was there a New Testament writer who was not a bishop?

According to the Catholic church, was there a New Testament writer who was not a bishop?

The only one that I am not sure of is St. Luke. Even here, I believe that he was a bishop, even if I can not find a source.

St. Peter, Prince of the Apostles was obviously a bishop. He was the the Bishop of Rome for approximately 25 years according to Jerome (De Viris Illustribus ch. I).

St. Paul was an apostle and bishop. The Catholic Church acknowledges St. Paul the Apostle as a consecrated Bishop during Apostolic times?

The Catholic Encyclopedia states that St. Paul actually consecrated Timothy a Bishop by St. Paul himself thus St. Paul had to have been a bishop.

From Scripture we learn that the Apostles appointed others by an external rite (imposition of hands), conferring inward grace. The fact that grace is ascribed immediately to the external rite, shows that Christ must have thus ordained. The fact that cheirontonein, cheirotonia, which meant electing by show of hands, had acquired the technical meaning of ordination by imposition of hands before the middle of the third century, shows that appointment to the various orders was made by that external rite. We read of the deacons, how the Apostles "praying, imposed hands upon them" (Acts 6:6). In 2 Timothy 1:6 St. Paul reminds Timothy that he was made a bishop by the imposition of St. Paul's hands (cf. 1 Timothy 4:4), and Timothy is exhorted to appoint presbyters by the same rite (1 Timothy 5:22; cf. Acts 13:3; 14:22). In the Third Clementine Homily (73), we read of the appointment of Zachæus as bishop by the imposition of Peter's hands. The word is used in its technical meaning by Clement of Alexandria (Stromata VI.13, 106; cf. Apostolic Consitutions II.32). "A priest lays on hands, but does not ordain" (cheirothetei ou cheirotonei) "Didasc. Syr.", IV; III, 10, 11, 20; Cornelius, "Ad Fabianum" in Eusebius, Church History VI.43.

The ordinary minister of the sacrament [of holy orders] is the bishop, who alone has this power in virtue of his ordination. Holy Scripture attributed the power to the Apostles and their successors (Acts 6:6; 16:22; 1 Timothy 5:22; 2 Timothy 1:6; Titus 1:5), and the Fathers and councils ascribe the power to the bishop exclusively. First Council of Nicaea (Canon 4) and Apostolic Constitutions VIII.28 — "A bishop lays on hands, ordains. . .a presbyter lays on hands, but does not ordain." - Holy Orders

St. John the Evangelist, St. Matthew the Evangelist and St. Jude Thaddeus were all bishops. Catholic tradition holds that all the Apostles were bishops.

An Apostle is a higher calling. An Apostle helps to lead the Church as a whole. A bishop leads an individual region.

The successors of the Apostles are bishops and not other apostles.

The authority of the Apostles proceeds from the office imposed upon them by Our Lord and is based on the very explicit sayings of Christ Himself. He will be with them all days to the end of ages (Matthew 28:20), give a sanction to their preaching (Mark 16:16), send them the "promise of the Father", "virtue from above" (Luke 24:49). The Acts of the Apostles and the Epistles of the New Testament show us the exercise of this authority. The Apostle makes laws (Acts 15:29; 1 Corinthians 7:12 sq.), teaches (Acts 2:37 and following), claims for his teaching that it should be received as the word of God (1 Thessalonians 2:13), punishes (Acts 5:1-11; 1 Corinthians 5:1-5), administers the sacred rites (Acts 6:1 sq.; 16:33; 20:11), provides successors (2 Timothy 1:6; Acts 14:22). In the modern theological terms the Apostle, besides the power of order, has a general power of jurisdiction and magisterium (teaching). The former embraces the power of making laws judging on religious matters, and enforcing obligations by means of suitable penalties. The latter includes the power of setting forth with authority Christ's doctrine. It is necessary to add here that an Apostle could receive new revealed truths in order to propose them to the Church. This, however, is something wholly personal to the Apostles. Authority and prerogatives of the apostles

Clement of Rome explicitly states that the apostles appointed bishops as successors and directed that these bishops should in turn appoint their own successors; given this, such leaders of the Church were not to be removed without cause and not in this way.

And thus preaching through countries and cities, they appointed the first fruits [of their labors], having first proved them by the Spirit, to be bishops and deacons of those who should afterwards believe. Nor was this any new thing, since indeed many ages before it was written concerning bishops and deacons…. Our apostles also knew, through our Lord Jesus Christ, that there would be strife on account of the office of the episcopate. For this reason, therefore, since they had obtained a perfect foreknowledge of this, they appointed those [ministers] already mentioned, and afterwards gave instructions that when these should fall asleep, other approved men should succeed them in their ministry. – Letter to the Corinthians 42:4–5; 44:1–2 (70 A.D). - Proof of Apostolic Succession that Goes Back to 70 AD!

St. Mark was the founding bishop of the See of Alexandria.

Believed to be the young man who ran away when Jesus was arrested (Mark 14:51-52), and the “John whose other name was Mark” (Acts 12:25). Disciple of Saint Peter the Apostle who travelled with him to Rome, and was referred to as “my son Mark” by the first Pope. Travelled with his cousin Saint Barnabas, and with Saint Paul through Cyprus. Evangelized in Alexandria, Egypt, established the Church there, served as its first bishop, and founded the first famous Christian school. Author of the earliest canonical Gospel. - Saint Mark the Evangelist

St. James the Lesser was the first Bishop of Jerusalem.

Cousin of Jesus. Brother of Saint Jude Thaddeus. Raised is a Jewish home of the time with all the training in Scripture and Law that was part of that life. Convert. One of the Twelve Apostles. One of the first to have visions of the risen Christ. First Bishop of Jerusalem. Met with Saint Paul the Apostle to work out Paul‘s plans for evangelization. Supported the position that Gentile converts did not have to obey all Jewish religious law, though he continued to observe it himself as part of his heritage, may have been a vegetarian. A just and apostolic man known for his prayer life and devotion to the poor. Martyr.

Having been beaten to death, a club almost immediately became his symbol. This led to his patronage of fullers and pharmacists, both of whom use clubs in their professions. He is reported to have spent so much time in prayer that his knees thickened, and looked like a camel’s. Soon after the Crucifixion, James said he would fast until Christ returned; the resurrected Jesus appeared to him, and fixed a meal for James Himself. - Saint James the Lesser

St. Luke was the author of Acts was the author of the Gospel that bares his name.

According to the early Church historian Eusebius, Saint Luke was born in Antioch of Syrian or Greek parents in 31 AD. As one of the first Gentile converts to Christianity, he became a travelling companion of Saint Paul on his many missionary journeys. Even after everyone else had deserted Paul, Luke was with him in his last days and final imprisonment in Rome. What happened to Luke after Paul’s martyrdom is not certain, but according to a fairly early and widespread tradition, Luke was unmarried and wrote his gospel in Greece at Boeotia, where he died at age 84 AD.

I can not find any clear confirmation one way or another whether St. Luke was a bishop. It is quite probable that the Evangelist Luke was in deed a bishop, but I can not find Catholic sources confirm this.

St. Luke is one of the most extensive writers of the New Testament. His Gospel is considerably longer than St. Matthew's, his two books are about as long as St. Paul's fourteen Epistles: and Acts exceeds in length the Seven Catholic Epistles and the Apocalypse. Thus it seems hard to fathom, that he was not a bishop, especially seeing it that he preached in Dalmatia!

According to the Orthodox tradition, St. Luke became Bishop of Thebes in Boeotia.

St. Luke was not a Jew. He is separated by St. Paul from those of the circumcision (Colossians 4:14), and his style proves that he was a Greek. Hence he cannot be identified with Lucius the prophet of Acts 13:1, nor with Lucius of Romans 16:21, who was cognatus of St. Paul. From this and the prologue of the Gospel it follows that Epiphanius errs when he calls him one of the Seventy Disciples; nor was he the companion of Cleophas in the journey to Emmaus after the Resurrection (as stated by Theophylact and the Greek Menologium). St. Luke had a great knowledge of the Septuagint and of things Jewish, which he acquired either as a Jewish proselyte (St. Jerome) or after he became a Christian, through his close intercourse with the Apostles and disciples. Besides Greek, he had many opportunities of acquiring Aramaic in his native Antioch, the capital of Syria. He was a physician by profession, and St. Paul calls him "the most dear physician" (Colossians 4:14). This avocation implied a liberal education, and his medical training is evidenced by his choice of medical language. - Biography of Saint Luke