Actually, the more accurate description would be Messianic Jews.
Again, I don’t see the term ‘messianic Jew’ anywhere in Scripture.
I do, however see Paul preaching that:
“All are alive in Christ Jesus”, indicating that the Church of Acts did not regard themselves as ‘messianic Jews’, but rather as followers of Christ, and, as Paul states, members of the Church. Significantly no distinction is made between Jews and Gentiles.