H4 visa holder (without EAD) as a Director in an Indian Company

Hi!

Can an H4 Visa holder (without an EAD) be a Director in an Indian company without any salary or monetary income?

Yes, as far as H4 is not actively employed and generating income in the US, you are good. If you generate any income outside of the US but you file your taxes in accordance with the US tax laws, you are good.