Obama urged voters to pressure Congress on the issue. "This effort is not over," he said, and change will come "so long as the American people don't give up on it."
Public support for tightening gun laws has dropped off as the Dec. 14 school shooting slips further into the past.
One month after the Newtown attack, 58 percent of Americans said they supported stricter gun laws, an AP-GfK poll found. This month, support was 49 percent.
Some specific gun proposals still have strong appeal, however, polling shows.
They're hugely popular: More than 8 in 10 Americans support requiring background checks for buyers at gun shows, according to the AP-GfK poll in January. So closing the "gun-show loophole" looked like a potential place where gun control Democrats and gun rights Republicans might agree.
Two senators, Patrick Toomey, R-Pa., and Joe Manchin, D-W.Va, tried to bridge the divide with a compromise that would subject buyers at gun shows and on the Internet to the checks but exempt noncommercial transactions like sales between friends and family. On Wednesday their measure was supported by a majority of senators but fell short of the 60 votes needed to advance.
The existing system, created under the 1993 Brady law, requires licensed gun dealers to submit a buyer's name before completing the sale. Convicted criminals and people who have been declared by a judge to be "mentally defective" are among those barred from buying a gun. Private gun owners and sellers at gun shows don't have to run the federal checks.
It's unclear how many buyers avoid scrutiny this way. Gun control advocates often claim that about 40 percent of guns are sold without the checks. But that's based on a survey from 20 years ago, when the background check system was just starting, and it was considered a rough approximation.
A few states have their own, more comprehensive background check requirements.
Another area of public agreement: Eight in 10 Americans want more done to prevent people who are mentally ill from buying a gun, according to Pew Research Center polling.
One way is to do a better job of getting treatment to people who need it. Obama wants to spend $235 million in federal money to identify and treat mental illness, especially in young people, and to study how to prevent shootings. The idea has appeal on both sides of the gun control debate, as well as among advocates for the mentally ill, although they stress that most people who need care aren't violent.
U.S. law bans gun sales to people who have been involuntarily committed or formally found to be dangerously mentally ill by a court or similar authority. But the federal background check system is weakened by paltry information from some states. Obama says he will do more to encourage states to share their mental health records.
A vote on a bipartisan proposal for improving mental health programs was set for Thursday in the Senate.
One of the most-discussed gun control ideas — reviving the 1994 ban on sales of "assault weapons" — couldn't get anywhere in Congress.
A majority of Americans — 55 percent — surveyed for the AP-GfK poll said they favored a ban on military-style, rapid-fire guns; about a third opposed it. Feelings run strong on both sides. Backers of gun rights are especially active lobbyists, however. These guns are popular with recreational shooters and people who consider them a menacing choice for home defense.
Other knocks against the proposal: Defining an "assault weapon" has always been tricky, and there's scant evidence that the old ban worked.
Under the now-expired law, some semi-automatic rifles and pistols were banned by name, including the Uzi, the AK-47 and the Colt AR-15, which is similar to the military's standard issue M16. Others were banned because they had a combination of characteristics listed in the law.