Stuff like this, while done with good intentions, can threaten to entrench discrimination by having different sets of rules and expectations for people based on their gender. A better solution would be to give 12 extra holiday days per year to everyone, but make it known that it's acceptable to take them under those circumstances.
Throwaway because no one wants to hear about periods. I am a woman and one time my body decided to have an extremely heavy and painful flow for about a week straight. I had been sick with a cold immediately before that, so I had already let my employer know that I was not feeling well and would be taking some days off, and I did not feel the need to tell my employer every little detail about what was going on. Every…
In Sweden for example, vacation days and sick days are separate, and up to a week, the employer is required to pay you during sick leave, and after that social security (equivalent) kicks in. This obviously doesn't use up your vacation days (where employers have no business in knowing what you're up to), but as for sick days it's not unreasonable to (have the option to) ask for validation for your absence.
I agree that employers (should) have no authority to know anything that's between you and your doctor, but I think it's also reasonable to have some verification (a note from doctor not specifying what the ailment is?) that you weren't just taking time off on the employers dime.