WASHINGTON—President Barack Obama on Wednesday said all gun buyers should be subject to background checks and that he would push "without delay" ideas for remolding the nation's mental health and gun laws, setting up a potential fight with gun-rights advocates.

"This time words must lead to action," Mr. Obama said from the White House, offering his most specific proposals yet in the wake of the shooting spree at a Connecticut school. "The fact that this problem is complex can no longer be an excuse for doing nothing."

Mr....