It is a private business. It has the right to refuse customers. If, in the opinion of the staff, she would not fit in the chair, or worse, damage it, then service should be refused. Two things:
1. Had she damaged the chair, who pays for a replacement? Do you think the customer would have ponied up the cash? No, she would not. And the spa would have been out the cost of the chair, and the lost revenue for the time the chair was out of commission. Those chairs aren't cheap.
2. Liability. If the customer has an accident and is injured by the chair for not being able to support her weight, would she have sued the establishment? Based on the crabbing she did on her Facebook page, it is reasonable to assume she would have gone after the business for financial compensation.
If it goes any farther than crabbing on Facebook, if I was the spa, I'd sue her into the stone age for impugning the reputation of my business. It's one thing to be angry because she was refused service. It's quite another if she slanders the spa in public.
It doesn't matter why she's fat. Whether she eats too much or has a glandular condition, the fact remains that serving her could cost the business more financially than they can afford.