There is a completely different set of rules for women, with a completely different set of origin stories.
But one of the rules is, that if a lady is wearing a men’s hat – such as a baseball cap – then the men’s hat rules apply.
As Musicat noted:
“… it’s not the presence or absence of head covering that indicates respect, but the adherence to custom.”
Custom also prescribes that you show respect for other nations’ national anthems.
There are several religious sects that would strongly oppose women having to uncover their heads in public. Mennonites and their offshoots come to mind.
Orthodox Jewish men often wear a yarmulke under their hat, so that they can take off their hat and show respect according to custom, without baring their head. So it’s not an insurmountable problem in the case of head covering. Maybe other religions have a problem with respecting flags or anthems, which would cause social problems.
I see you don’t question why you “have to” stand, which is also a customary way of showing respect during the playing of a national anthem. Is it because you don’t mind standing? Or is it because you just hate to part with your hat, even for a few minutes?
for edification only - regarding “boorishly oblivious” and"outdated persnicketyness…"
If one is taught that a certain behavior is an indication of respect or good manners, clinging to those behaviors is almost unavoidable, even if the times have (rapidly) changed. To wit: I am an old guy. When I enter a store or an elevator, for instance, I feel compelled to remove my hat - even if it’s a nasty old watch cap. It makes me feel as if I’m doing the right thing. I am painfully aware of the fact that I, and my friends, are dinosaurs and that the sooner we just shuffle out of here and get out of the way, the better. The backward baseball hat is an excellent symbol of the evolution of sensibilities. But keeping my hat on in those situations feels wrong, even if logic and current thinking say it isn’t. Be patient, we’ll be gone soon with our outdated persnicketyness and you will have not have to deal with our demonstrating such things as meaningless customs that merely show regard for others.
It’s often been the case in the past that a woman taking off her hat has to deal with more details than a man. Hat pins, for instance, or hair that’s arranged just so and will fall if the hat’s removed.
Oh, those poor silly helpless girls. It would be sooooooo mean to monkey with their vanity. Hey, if taking off my cap will make my hair look goofy or expose my bald spot maybe I can beg an exemption as well. No? Cuz that would make me a sissy?
I categorically reject any custom that demands different arbitrary superficial gestures from men and women.
Did someone mention Don Draper in the elevator? The same Don Draper who spends his life as a philandering, sexist asshole? He wasn’t enforcing morality inthat elevator. He was just swinging his dick. What does custom say about humiliating another man in public and using your physical strength as a threat to make him conform to your personal tastes?
You can stand. You can put your hand over your heart. You can get down on your knees. You can do that thing where you put your palms together and bow. There are all sorts of signs of respect that are the same for all genders.
And if you’ve been reading, it hasn’t been demanded.
(Just genuine curiosity CC), isn’t a “store” a public place where one would wear a hat? Does a mall (an indoors one) change things? Are the hallways public but the stores not? Hat removal in elevators confuse me too – unless there are ladies in the elevator… in which case I guess the “company of ladies” rule applies due to the confining space.
(I’ve never worn hats other than for medieval re-enactment and never learned the hat ettiquette rules, so it’s all fascinating and a bit mystifying).
So if I (a New Zealander) am in the US and the US national anthem is played I should stand, uncover… and put my hand over my heart? (Actually I would / did stand, and don’t wear a hat… but the hand on heart seems wrong – 'course I also happily sang your anthem… 'cos it’s cool). 
The law quoted above says “should” - are there any penalties if it ever came before a judge? I doubt it.
pats Musicat consolingly
There, there
It simply could not come before any competent judge under a competent prosecutor. That provision of the U.S. Code that is essentially a nullity. As I said before, if anyone tried to enforce it, it would immediately be struck down as a violation of the constitution.
I think that just standing is what’s prescribed. Hand-on-heart – a salute – would be incorrect for someone not a citizen of the country whose anthem is being played.
The general idea is to show respect for others’ national practices, but not to participate in those not your own.
For some reason, hats seem to generate a lot of arcane tradition. These days, the orientation and angling of the bill of a baseball cap has been added to list of stuff to know. But all the preceding custom is still out there.
You sir are a gentleman of which I wish there were more. I’m no prude or super patriot but it’s a sad day when it’s become too much of an imposition for someone to show a little freakin’ decorum.
If your hat is attached to your hair with pins, you’ve got a legitimate excuse to not whip it off when they play the anthem. This particular reason for the difference isn’t about vanity so much as practicality. Not that this is the only reason that men are expected to take their hats off and women aren’t, but it’s part of it.
It’s much more practical to let everyone decide individually what to do with their hats and to keep your sartorial expectations of others on hold.
Just to clarify, my “should I stand, uncover…” was more a question of etiquette than legal compliance… the quoted text didn’t seem limited to US citizens (though the quote is only a fragment).
Removing one’s hat during the national anthem is an American custom. Customs are an important part of almost every society, passing on and cementing societal values and principles. It might be worth mulling over which customs deserve to be retained and which should fall by the wayside, but that thread would be more appropriate in IMHO. Actually, hasn’t this thread morphed into IMHO fodder anyway?
See if you can get your teenage children to believe that. One way or another people judge each other by their clothing.
And perhaps a sign of a maturing society is that we try to teach people away from their baser instincts. Tribalism and racism is also naturally ingrained in human nature, as is magical thinking. We teach away from that.