Is it healthy to wash hair everyday using only water and no shampoo?

Healthy but unusual: I guess there is no special health issues, depends how oily your skin is and how much environmental pollution you are exposed to. Any skin or hair needs some soap or shampoo to wash away certain dirts .As you wash your hands and body with soap, you need to wash your head as well with a shampoo.

Health vs hygiene: Many people of the third world countries don't have shampoo and so often washing without soap/shampoo. Scalp/hair follicles actually produce the oil to protect the hair and we use shampoo to wash that off, then use conditioner to protect it (ironic?)--it is so done for hygiene reason, not health. Unwashed hair is actually (often) healthier hair (may not be socially nice, but healthy). Good luck.