WASHINGTON—Sen. Susan Collins, who has clashed with Republican leaders over taxes and health care, on Friday played the role they needed from her: A Republican woman pledging the decisive vote to elevate Judge Brett Kavanaugh to the Supreme Court.

“I will vote to confirm Judge Kavanaugh,” she said at the end of a 45-minute speech laying out her rationale—starting with his judicial opinions, continuing through her thinking about his rights to a fair process, and weighed against accusations from California research psychologist...