Mr. Obama voiced irritation over the issue.

“Each time this comes up,” Mr. Obama said in his speech, “we are fed the excuse that common-sense reforms like background checks might not have stopped the last massacre, or the one before that, or the one before that, so why bother trying. I reject that thinking. We know we can’t stop every act of violence, every act of evil in the world. But maybe we could try to stop one act of evil, one act of violence.”

Nearly 21 million gun sales were processed through the background check system in 2014, but some industry analysts say as many as 40 percent more firearms could have been sold through private transactions not subject to background checks. Even the most hopeful advocates say the new plan would affect only thousands of sales.

Proposals that would have the biggest effect have long been shelved by even the most ardent gun control advocates who now see an assault weapons ban or mandatory gun buyback programs like ones in Australia in 1996 and 2003 as political fantasy.

Modest as the new measures may prove to be, the response was unrestrained. Republican presidential candidates and congressional leaders greeted them with peals of protests and angry claims of a “gun grab” that would violate Second Amendment rights. Gun control advocates hailed them as a breakthrough in what has often been a losing battle to toughen firearms restrictions.

The families of gun victims and gun control activists crowded into the White House and watched Mr. Obama break down as he recalled the young children gunned down by an assailant in 2012 at Sandy Hook Elementary School in Connecticut.

“First graders,” the president said, his eyes drifting off and becoming red with tears. He wiped his face and paused to regain his composure: “Every time I think about those kids, it gets me mad.”