But the Utah Republican didn't support more general gun control measures that Rep. Donna Edwards, D-Md., called for. She suggested reinstituting an assault weapons ban and requiring background checks on weapons purchased at gun shows or from an individual.
"We've got to get the guns. We just do," she said.
Chaffetz said in the immediate aftermath of the shooting there has been "a lot of conjecture out there that I don't think will necessarily solve this particular problem."
"I'm a concealed carry permit holder. I own a Glock 23. I've got a shotgun. I'm not the person you need to worry about and there are millions of Americans who deal with this properly," he said. "But we have to look at the mental health access."
President Barack Obama has said it is time to take "meaningful action" to stop or at least reduce mass shootings, though he has not specified what that might be. In the second presidential debate, earlier this year, Obama did mention limited access to weapons by the mentally ill.
Sen. Richard Blumenthal, D-Conn., also appeared on This Week, and said he's prepared to push for new gun control measures, though he wants to delay a detailed discussion of his ideas "out of respect for the families and their grief."
A former state attorney general, Blumenthal said: "We need to do something at the very least about the high-capacity magazines used in this crime."
Beyond exploring changes to the mental health system or debating gun control, Chaffetz and Edwards agreed that the nation must confront "a culture of violence" that permeates video games and movies.
"I think the movie ratings are terribly misleading when it comes to violence, death and gore and glamorizing it," said Chaffetz. "We are also going to have to look at families and communities and churches. It is not just a government solution."