Does childbirth count as a medical expense you can deduct?

Absolutely!

Medical expenses are the costs of diagnosis, cure, mitigation, treatment, or prevention of disease, and the costs for treatments affecting any part or function of the body. These expenses include payments for legal medical services rendered by physicians, surgeons, dentists, and other medical practitioners. They include the costs of equipment, supplies, and diagnostic devices needed for these purposes.

Yes. any out of pocket expense not covered by your health insurance.