Texas Sen. John Cornyn (R) on Sunday said that he opposed a bill to close the so-called gun show loophole and expand background checks to Internet gun sales because only better mental health laws will ensure that the Newtown mass shooting victims "did not die in vain."

"In my meeting with the Sandy Hook families, they told me that -- and of course, who wouldn't have sympathy and empathy for these people who have suffered a terrible loss -- but what they told me is that they wanted to make sure their loved one did not die in vain, that something good would come out of this," Cornyn told Fox News host Chris Wallace. "And so I think -- that's why I'm focused like a laser on the mental health component."

"But forgive me, sir," Wallace interrupted. "They are focused on tougher gun control. Specifically, the background check."

"Well for example, [Newtown shooter] Adam Lanza stole his mother's guns," Cornyn explained. "A background check would not have stopped that problem, that incident. A background check should have stopped James Holmes in Tucson, it should have stopped the Virginia Tech shooter."

"In other words, I think the mental health issue is the common element that we ought to be focused on, and I think we can do some good things," the Texas Republican added. "But I'm not for symbolism over substance. I think we can't just pat ourselves on the back and say we're going to pass some enhanced penalties for trafficking or other issues or background checks when they don't really go to solve the problems that cause these terrible tragedies."

Cornyn pointed out that the bipartisan legislation proposed by Sens. Joe Manchin (D-WV) and Pat Toomey (R-PA) would not have prevented any of the four most recent mass shootings.

"The [Newtown] parents say that doesn't matter," Wallace noted.

"Well, what matters to me is that we not just engage in a symbolic act and pat ourselves on the back and say we've done something good and left the problem unsolved," Cornyn insisted. "I would like to try to solve the problem by focusing on the common element of these recent tragedies, which is the mental health issue."

Watch this video from Fox News' Fox News Sunday, broadcast April 14, 2013.