Cornyn said he agreed with Obama that Wednesday had been a shameful day but added it was because of the president's own comments, rather than the events on the Senate floor.
"He could have taken the high road...instead he chose to take the low road, and I agree with him it was a truly shameful day."
Reid offered no timetable for renewing the drive to enact legislation that Obama has placed near the top of his domestic agenda.
Another Democrat, Sen. Joe Manchin of West Virginia, said the proposed expansion of background checks that he co-authored would have passed easily had it not been for the National Rifle Association's decision to take the vote into account in deciding which candidates to support or oppose in 2014.
"If they hadn't scored it, we'd have had 70 votes," he said. Instead, it drew 54, six short of the 60 needed to advance.
Manchin also told reporters at a breakfast sponsored by the Wall Street Journal that the outcome would have been different if the Senate had acted more quickly after the shootings at Sandy Hook Elementary School in Newtown. "If we'd have gone to a bill like this immediately, boom," he said, predicting it would have received 65-70 votes.
Later, in the Capitol, he renewed his accusations that the NRA had spread lies about the proposal he had hoped to pass, and that fellow lawmakers had evidently believed them. "Reading is a lost art," he said.
Obama spoke in clipped, angry tones at the White House on Wednesday after the Senate scuttled legislation he had campaign for energetically.
"I see this as just Round One," the president said, flanked by relatives of Newtown's victims and former Rep. Gabrielle Giffords, who was shot in the head in Tucson, Ariz., in 2011.
Looking ahead to the 2014 congressional elections, he added, "If this Congress refuses to listen to the American people and pass common-sense gun legislation, then the real impact is going to have to come from the voters."
Obama blamed lawmakers' fear that "the gun lobby would spend a lot of money" and accuse them of opposing the Second Amendment's right to bear arms.
A spokesman, Josh earnest, told reporters on Thursday, "we're pretty close to a consensus on this just about everywhere except in the United States Congress. And as the President alluded to yesterday, I think that is an indication of the pernicious influence that some special interests have in the United States Congress. And that is going to require a vocalization of public opinion to overcome it."
Emotions were high on Wednesday at the Capitol.
When the background check amendment failed in the Senate, Patricia Maisch, watching from a visitors' gallery, shouted "Shame on you!" Maisch helped restrain the gunman at the 2011 Tucson shooting in which six people died and 13, including Giffords, were wounded.