What I hear to you say is that you were fired because management thought you treated a customer in an insulting way, but you didn't do it, some other employee did it, and you both got fired. That sounds very unfair to me. But it doesn't sound unlawful. Generally speaking, an employer doesn't have to be RIGHT when it thinks an employee did something wrong. People innocent of retail theft get fired all the time when the employer thinks they committed retail theft. It's not UNLAWFUL to fire an at-will employee because you wrongly think he insulted a customer. It's just UNFAIR
This whole response assumes you were an at-will employee. At-will employees can be fired for any reason or no reason at all, unlawful discrimination excepted. I don't hear any unlawful reason for your firing in what you tell us. So, as far as I can tell, your remedy is to look for another job.
That's not legal advice as I'm not licensed in Georgia. It's just my take on the facts you presented above in light of general principles of law. If you need legal advice, please consult a lawyer who holds Georgia licensure.
Good luck in your job search.
I agree with Mr. Taylor's answer. An "at will" employee can be fired for any reason, or no reason at all, as long as the employee is not fired or discriminated against on the basis of race, color, national origin, gender, age, religion, or disability. It may be unfair to be fired for something you didn't do, but it's not illegal.