TAMPA, Fla.—Republicans kicked off their national convention Tuesday with a day of speeches aimed at bolstering a weakness in Mitt Romney's bid for the White House—his tepid support among women voters—while building the argument for a policy platform that calls for federal spending cuts and a Medicare overhaul.

On a day that saw convention delegates hold the roll-call vote to make Mitt Romney the party's presidential nominee, Republicans gave a prime-time speaking role to Ann...