First, that's tough, sdoshier, and I'm sorry you have to deal with it.
In terms of the Americans with Disabilities Act* and what responsibilities your employer has to you, I'm afraid that you don't have a lot of recourse. You are protected if you can fulfill the "essential functions" of the job. If you are unable to speak, then you can't perform the essential function of this job.
Are there any jobs available at the call center that are less stressful? It sounds like it's a pretty general thing.
Also, are there any options for text-based communication? Online customer service, replying to emails, that sort of thing?
If there aren't, then I think your only option is to focus on mastering the stutter, or find a new job. Have you had any luck thus far with trying to overcome the stutter with training?
*This is if you're in America, but you might not be.