WASHINGTON—On the eve of a critical Senate hearing that could decide the fate of Supreme Court nominee Brett Kavanaugh, a third woman came forward to accuse him of sexual misconduct in the early 1980s as partisan anger deepened on both sides.

The revelations shocked Capitol Hill, where Republican lawmakers are seeking to advance a nomination that legal scholars and lawmakers have said could cement a conservative majority.

Judge Kavanaugh’s judicial elevation, once expected to move smoothly through the Senate, now hangs on the votes of a handful of GOP senators who said they were troubled by the late-breaking claims. Sens. Jeff Flake of Arizona, Susan Collins of Maine and Lisa Murkowski of Alaska all said they would determine their votes after Thursday’s Senate Judiciary Committee hearing.

With a 51-49 advantage, Republicans can’t afford more than one defection, assuming no Democrat votes for him.

At the hearing, Judge Kavanaugh and Christine Blasey Ford, who has accused him of sexually assaulting her when they were teenagers, are scheduled to testify separately about the alleged incident. The judge has denied Dr. Ford’s claims.