Another Republican, former Education Secretary William Bennett, made a similar argument, saying the political debate should be put on hold while emotions are still high.
"The whole nation is mourning. It's an important moment. Let the tears dry before we head off into all these directions at once," he said on NBC's "Meet the Press."
Bennett also agreed with the idea that schools should have a gun.
"I'm not so sure I wouldn't want one person in a school armed, ready for this kind of thing," he said. "It would have to be someone who's trained, someone who's responsible, but my God, if you can prevent this kind of thing."
Polls have shown that the public remains divided on the gun laws. A CNN/ORC International survey conducted in August -- shortly after the Aurora, Colorado, movie theater shooting and another one at a Sikh temple in Wisconsin - found that 50% of Americans favor no restrictions or only minor restrictions on owning guns, while 48% support major restrictions or a complete ban on gun ownership by individuals except police and other authorized personnel.
Those numbers are identical to where they were in 2011, and the number who support major restrictions or a complete ban has remained in the 48%-to-50% range for more than a decade.
Though their differing opinions in the debate may be sharp, Republican and Democratic politicians all agreed on one thing Sunday: No single piece of legislation will be able to stop the violence completely. As long as there's a will and an unstable mind, there's a way, they said.
Malloy illustrated that point, telling CNN the gunman in Friday's shooting literally "shot his way into the building," breaking past the school's security system.
But retiring Sen. Joe Lieberman, an independent from Connecticut, said on Fox that "the stronger our gun-control laws are, the fewer acts of violence - including mass violence - will happen in our society."
Others emphasized an additional need to boost mental health programs in the country. Democratic Gov. John Hickenlooper of Colorado, who was intimately involved in the aftermath of the Aurora movie theater shooting, said Colorado has spent almost $20 million in new programs to support those dealing with mental illness.
"That's something we can do immediately without getting into some of the battles of gun legalization or restricting access to guns," he said on CNN, though acknowledging some gun laws need to be tweaked.
In particular, he said the debate should focus on access to high-capacity magazines. His support for tougher laws in the state marks a change in policy for the governor, who earlier this year said stricter gun laws would not have helped.
Still, Hickenlooper argued the "country is based on the Second Amendment."
"My grandfather taught me how to shoot and clean a 12-gauge shotgun and showed me how to hunt, and I've showed my son," he said. "That tradition is very powerful throughout this country."
Rep. Jason Chaffetz of Utah said he legally owns a shotgun and a Glock, but he's "not the person you need to worry about."
"There are millions of Americans who deal with this properly. It's our Second Amendment right to do so," the Republican congressman said on ABC's "This Week." "But we have to look at the mental health access that these people have."
While a debate over gun rights quickly sparked after the Aurora tragedy, it wasn't long before the conversation began to fade, as a presidential election squarely focused on the economy soon dominated national dialogue.
But Sen.-elect Chris Murphy of Connecticut said Americans should not expect the newly resurfaced debate to go away anytime soon.
"Frankly the tipping point should have happened a long time ago, but if this is the tipping point, then we're going to go down to Washington and prompt a conversation that's long overdue," Murphy, a Democrat, told CNN chief political correspondent Candy Crowley.
Sitting next to fellow Democrat Connecticut Sen. Richard Blumenthal, Murphy recalled a certain plea that the elected officials encountered earlier Sunday in Newtown.
"A young man grabbed us in a church this morning, sobbing, and said 'Don't let his happen again.'"