WASHINGTON—A Republican senator who party leaders feared might join Democrats in seeking new testimony said Thursday night he will vote against calling witnesses, bolstering the GOP’s chances of bringing the impeachment trial of President Trump to a rapid conclusion.

Sen. Lamar Alexander (R., Tenn.) announced his plan after senators wrapped their second and final day of questioning Mr. Trump’s legal team and House Democratic impeachment managers, during which the two sides sparred over the limits of executive and congressional...