I work with the public, too, and I can definitely sympathize with you that customers can really suck. The slightest misunderstanding can turn into a complaint and because companies are so concerned about overall customer satisfaction, they have to react to the complaints. Regardless of whether you were right or wrong, they're very concerned over what the customer's overall impression was.
I wouldn't let it get to you. If you see the customer again, be polite and super helpful. Only apologize again if your boss tells you to.