WASHINGTON—Judge Brett Kavanaugh’s nomination to the Supreme Court was all but secured on Friday when Sen. Susan Collins declared her support on the Senate floor, capping a tumultuous 13-week battle that was stirred in recent weeks by allegations of sexual misconduct and raw emotion in both parties.

The day unfolded with slow suspense, after Judge Kavanaugh cleared a key procedural hurdle in a 51-49 vote, but the handful of undecided senators announced their final stances one at a time over the course of the afternoon. By...