Is it possible for fibromyalgia to be healed holistically?

Possibly. Anything that reduces stress or the effect of stress or that produces deep (dreaming) sleep may help.
I believe so. If by holistic you mean integrating the body, mind and spirit to overcome fibromyalgia then the answer is a wholehearted yes.
Nutrition. Absolutely, there are many "holistic" or alternative treatments for fibromyalgia. Acupuncture is proven by research to improve fibromyalgia symptoms and specific dietary modification can also be helpful in lowering inflammation that can increase symptoms. Autoimmune disorders, candida, and hormone imbalance are also thought to be contributing factors that when manage can alleviate the condition.
Yes. Functional medicine can bring you back to wellness. I would look at your GI tract and check your stool for bad bacteria, leaky gut syndrome and food sensitivities. There are several other tests I would do also.
POSSIBLY. Several doctors and laypersons have written books supporting assertions that fibromyalgia can be healed with natural approaches. Find through amazon. Several clinical reports tout aquatherapy. If doctors readily admit cause is unknown, assertions that certain natural practices (based on age-old therapies; worthy empirical evidence) are "not possible" derives from unseemly medical doctor hubris.
Absolutely not. There is no controlled data to suport any benefit from holistic approaches in the treatment of fibromyalgia.