The earliest references of Acts by the early church unanimously indicate that Luke, the companion of Paul, was the author. Internal evidence, while not conclusive, does nothing to contradict this belief. Scholars are united in their view that the same person wrote both Luke and Acts (Bruce, 2). The vocabulary and style of both writings is the same (Hawkins, 16-29, 174-89). The concerns of both writings are parallel: e.g., concern for those outside of Judaism, the prominence of women, and the role of the Holy Spirit.
The “we” sections of Acts (16:10-17; 20:5-21:18; 27:1-28:16) suggest that the writer was a companion of Paul. There is no means, however, to prove or disprove conclusively that this companion was Luke.
Several scholars have commented on the medical language in Luke-Acts, suggesting that there is a higher incidence of precise medical terminology used in the description of ailments and conditions than in the rest of the NT (Hobart: Harnack, Luke the Physician). This might point to Luke, the beloved physician (Col 4:14). More careful scholarship, however, has called into question the strength of this observation (Cadbury).
The author of Acts probably was Luke, the companion of Paul, as the early church attested. However, this cannot be proved conclusively on the basis of the available evidence.