Did the Council of Carthage select only the books of the New Testament for the canon?

Your friend is wrong about the Council of Carthage. Canon 36 reads:

[It has been decided] that nothing except the canonical Scriptures should be read in the Church under the name of the divine Scriptures. But the canonical Scriptures are: Genesis, Exodus, Leviticus, Numbers, Deuteronomy, Joshua, Ruth, four books of Kings, Paralipomenon two books, Job, the Psalter of David, five books of Solomon, twelve books of the Prophets, Isaiah, Jeremiah, Daniel, Ezechiel, Tobit, Judith, Esther, two books of Ezra, two books of the Maccabees. Moreover, of the New Testament: Four books of the Gospels, the Acts of the Apostles one book, thirteen epistles of Paul the apostle, one of the same to the Hebrews, two of Peter, three of John, one of James, one of Jude, the Apocalypse of John.

Thus [it has been decided] that the Church beyond the sea may be consulted regarding the confirmation of that canon; also that it be permitted to read the sufferings of the martyrs, when their anniversary days are celebrated. (From Denzinger’s Enchiridion Symbolorum, translated and published in English as The Sources of Catholic Dogma)

Two key points should be noted. First, while the names and divisions of some Old Testament books differ from contemporary usage (for example, the four books of Kings are, in modern Bibles, divided into 1 and 2 Samuel and 1 and 2 Kings), the canon is that of the Catholic Bible, not of the Protestant. Second, this canon was to be confirmed by the “Church beyond the sea”–which means Rome.