WASHINGTON—President Barack Obama, seeking to restore confidence in his leadership, declared in his State of the Union address Tuesday that he would use executive power to try to narrow the gap between rich and poor and speed the nation's economic recovery.

Mr. Obama's speech was essentially a manifesto designed to inject new vigor into his languishing agenda and guide his presidency through the partisan divide in the capital. The goal was to position the president as the champion of struggling Americans fed up with the bickering...