No, they cannot. You should speak with a local attorney regarding your options for educating the business community. Your local and state bar associations can give you a referral.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.
Denying service to a customer because of the breed of the service dog impinges upon the Americans with Disabilities Act.
As Attorney McFarland-Taylor suggests, an attorney can help educate the establishment that denied you service as to the consequences of its actions.
The foregoing is for general information purposes and does not establish an attorney-client relationship.
This is a good question. I assume by "fully registered" you mean you didn't just register from some website that provides a "license" for a dog that gives "emotional support" or something similar. Those registrations don't qualify under federal law. I'll also assume that by "disabled," you mean you have a disability defined or qualifying under the Americans with Disabilities Act (the ADA). Lastly, I'll assume that your local area doesn't have any breed-specific legislation prohibiting "vicious" animals or pit bulls (not the same thing) from businesses.
As a general rule, the ADA forces businesses to allow service dogs (and miniature horses) on the premises. Fear of the dog is not a valid reason to exclude it. The courts haven't ruled on whether breed-specific legislation is a reason to exclude a service dog.
So, if my assumptions are correct, the business had no right to exclude your service dog, and your rights under the ADA may have been violated. You should speak to an ADA lawyer.
The contents of this answer should be considered friendly advice, not legal advice (I'm a pretty friendly guy), and the answer should not be construed to constitute an attorney-client relationship. If you'd like actual legal advice, call me for a free consultation at 813-635-0222. Also, if you liked this answer as much as my big ego thinks you did, be sure to click the thumbs-up button!