Putting Government in a position of power over the Family
In Catholic teaching, we have a thing called subsidiarity. I've spoken about it before. Basically there are tiers to society with the family being the lowest and the federal government the highest. A tier at the bottom is supposed to handle the most things and what it can't it gets to move up in sphere. So the federal government is in charge of defending the country because a family would lack the resources and man power to do such a thing.
In the case of regulating one's speech, I believe that the family is the best fit for this. My own family is a great example of this because there are members of my family who have no problem with racist speech. Despite being Christians (although not Catholics), I have a few people in my mother's family who say some rather nasty things. My mother has in fact done some policing of this matter. She's pretty much called them out on this and has told them that under know circumstances is it appropriate to talk that way. I don't think that my family members need to be have some law in place telling them to stop speaking racist language. What is most effective and leads to less problems is other family members saying it's inappropriate or co-workers or friends. Peer pressure in other words.
What would happen if we allowed the government to regulate speech?
Well in other countries there are restraints on speech. In the extreme there are blasphemy laws which protect Muslims and often Christians are the target of use. Restraining speech also often restricts a Christian's ability to evangelize, which is something we are all called to do. It never goes well when a country restricts speech because it is difficult to draw a line in the sand. Historically countries have never been able to regulate it without restricting the rights of minority voices including Christians.
Which leads me to my next point...
The Church doesn't believe that everything immoral should be legislated.
God allowed free will knowing that even though it was a good thing, bad things would happen as a result. Free speech, in my mind, is like that. Having the ability to evangelize, worship, protest immoral acts, etc are all available to Christians in the US because of free speech. All other acts of evil happen in spite of free speech.
Moreover the Church doesn't believe in legislating morality. St. Thomas Aquinas actually has a test for whether something should be legislated and something should not be. I believe legislating speech is something St. Thomas Aquinas would frown upon.
Bishop Robert Barron actually talked about legislating morality in an interview on the Rubin Report. He drew criticism for saying that gay marriage is not necessarily a front to fight on, but his point that we should not legislate everything that is immoral is still a valid one. We can of course disagree about what things should and should not be legislated, but the fact remains that the Church teaches that not everything should or has to be.
So there you go, something to think about next time the argument is made that something should be legislated.