As Congress begins debate on authorizing a new war, senators voted Thursday to install Ashton Carter as the new secretary of defense, making him the top civilian official overseeing the fight against the Islamic State while also trying to grapple with Iran, Russia and other hotspots amid tight Pentagon budgets.

Republicans said they hoped Mr. Carter, who has previously served as the No. 2 civilian at the Pentagon, would be an independent voice willing to push President Obama beyond his reticent approach — and the early going suggests they are right.