No. And whether they do or don't does not effect society's perception of them in any recognizably negative way.
Well it's basically, "Comply with our arbitrary expectations or face discrimination." I don't know what it's like in other countries, but in the US it's expected. There's not really a "choice" at all.Women might face more external pressure, but ultimately it's their own choice to either do it or accept the possible consequences of skipping it.
I grew up in the US and think completely shaven women are as hot as it gets. But I realize that that's a product of being socialized to expect certain things. You can't really call it a fair choice until both options are accepted in society.