Think your friend is correct. At this point, it's truly your problem- your the one worked up about it at this time. What he said was wrong. And the decision to let him go or keep him should be based on what is best for your business. If you are truly worried he will give off a bad image, do it. If your upset because he was acting like an idiot temporarily, you may not be thinking with your head and more with emotions. (I usually regret those decisions)
No matter what, you should have a conversation and let him know that it isn't acceptable and why. I would suggest an employee handbook be made if you don't have one. You can let everyone know what is okay and what is not from the start. In the end, it has been worth the time and trouble more than once for me.
This may be a situation where you can come out ahead if you take the high road and let him know it is a warning and why. People say stupid things all the time that they later regret. Forgiveness with good timing, may do far more good than a poke with a stick.
My wife and I had a conversation last week. She said I didn't completely understand, I've never been followed around the store because of the color of my skin. I told her that she is way to aware of her skin color. Try walking into a rolex store no matter what color you are with dirt on your hands, mud on your boots and patched jeans- someone is going to see if you need assistance finding the nearest exit. People are divided on something in this country on a daily basis. Skin color is only one aspect. When I was growing up, my family was not well off. I went to school with a bunch of doctors and lawyers kids. There was a difference and at times, it was definately pointed out and I was treated differently. We have all been treated differently for one thing or another. Intelligence/ignorarance, athletic ability/inability, pretty/uggly, rich/poor, religious/denominational, political views.... you name it. I now probably make more than most of those kids I went to school with. They were just trying to tear someone else down to make themselves feel better. It tells you what they think of themselves more than it does about you.
That is my peace on the subject matter.