Did Constantine or the Council of Nicaea choose the books of the Bible?

Eusebius of Caesarea c. 260–c. 339 ADChurch History 3.25.1–6

First we must place the holy four of the Gospels, followed by the Acts of the Apostles. After this, the letters of Paul; then the first letter of John and likewise the first of Peter must be accepted. After these, if it seems right, the Revelation of John. These are the recognized books. Among the disputed books, which are nonetheless known to most, are the letter called James's, and Jude's, the second of Peter, and the second and third of John. Among the spurious are the Acts of Paul, the Shepherd, the Revelation of Peter, the letter of Barnabas and the so-called Teachings of the Apostles. ... We have made this list so that we may know these books, and also those that heretics put forward under the apostles' names, such as the gospels of Peter, Thomas and Matthias, which no churchman in the succession has ever thought fit to mention in writing.

Written in the years around Nicaea. Eusebius describes the canon as the churches already received it, with a few books still debated. He does not mention any council deciding.

Constantine (letter in Eusebius, Life of Constantine) c. 272–337 ADLife of Constantine 4.36

It has seemed right to tell your wisdom that you should order fifty copies, on prepared parchment, easy to read and convenient to carry for use, to be written by skilled scribes who know their craft exactly: copies, that is, of the divine Scriptures, whose preparation and use you know to be most needed for the instruction of the Church.

Constantine's whole role in Scripture, as far as the sources go: he paid for copies for the new churches of Constantinople, about seven years after Nicaea, and left the contents to Bishop Eusebius.

Council of LaodiceaCanons 60

These are the books of the New Covenant: four Gospels, according to Matthew, according to Mark, according to Luke, according to John; the Acts of the Apostles; seven catholic letters, one of James, two of Peter, three of John, one of Jude; fourteen letters of Paul.

About forty years after Nicaea, local councils were still writing their own lists. They do not cite Nicaea.

Jerome c. 340/345–420 ADPreface to Judith Preface

Among the Hebrews the book of Judith is read among the apocrypha, and its authority is judged less fit to settle disputed points. ... But because the Nicene Synod is read to have counted this book among the sacred Scriptures, I have agreed to your request.

The only ancient claim that Nicaea said anything about a biblical book. No such statement survives in the council's canons or creed. Jerome may mean that the bishops quoted Judith.