It seems likely that a majority of the Justices will indeed conclude that there is no constitutionally protected right to get an abortion, because the Constitution "makes no mention" of abortion, and doesn't otherwise implicitly authorize.
But the Constitution does make mention of the free exercise of religion. In Fulton v. City of Philadelphia (2021), six Justices (Thomas, Alito, and Gorsuch, plus Breyer, Kavanaugh, and Barrett) took the view that the Free Exercise Clause provide at least some sorts of religious exemptions from neutral, generally applicable laws. Three of them (Thomas, Alito, and Gorsuch) took the view that such exemptions have to be given unless denying the exemption is narrowly tailored to a compelling government interest. And at least five Justices (Thomas, Alito, Gorsuch, Kavanaugh, and Barrett) have taken the view that such a "strict scrutiny" level of protection for religious objectors should apply if the laws have some secular exemptions. All of them took the view that "free exercise of religion" doesn't just apply to worship services or religious rituals, but also applies to people's presumptively right to act consistently with their religious beliefs (or to refuse to act when the action would violate their religious beliefs), even when the action or inaction is illegal.
So this brings back an old question, but one that had been not very important so long as Roe and Casey generally protected abortion rights: When are abortions protected by the Free Exercise Clause, even if not through the Due Process Clause?
