He is absolutely, 100% wrong. It may be true for mass shootings, but not for overall gun violence.
A lot of gun deaths are suicides. Those are mental illness.
But there are categories that are just idiots with guns, like people who get into fights outside bars.
There are also a surprising number of men who murder a woman who is pregnant with their child because they don't want the baby to be born.
Sometimes murder is just murder and there is no diagnosis. Not all aberrations in society are mental illness.