This is kind of a part two to my earlier post office description, because it didn't fit well in the first entry. On one wall of the post office was an notice telling men to sign up for the draft, called "selective military service." Canada doesn't have that, and I didn't know the US had that still. You can ask for a form here to sign up, and I'd like to see it, but being foreign and female I'm afraid I would be regarded with suspicion for such a request.
The USA only drafts men into their armed forces, but accepts applications from female volunteers. That is probably a sore point with some men, and would certainly become more so if the US ran out of military volunteers and started calling up those registered for the draft. I imagine that will have to be redone at some point, but what a can of worms!
Under the laws of the US, the draft should probably be gender-blind. Women are supposed to be treated the same as men in voting, running for public office, receiving government benefits and every other dealing with government, so it's not exactly fair to the men for women to have a special exception for warfare. It's also, in a perverse way, unfair to the women, because if men are asked to pay a higher price for citizenship that justifies them receiving preferential treatment.
One reason women don't traditionally go to war is that they have not been considered capable of the duties. While being big and strong and testosterone-charged is very important when it comes to beating your opponent to death with a stone club, its importance shrank with the introduction of firearms, and size became a disadvantage in some military professions with the advent of tanks and fighter jets and other mechanized vehicles. Given training and indoctrination and assignment to appropriate war trades, I think women are as capable as men at waging war.
The things that women can do that men can't make a different and more valid argument against women fighting in wars. In order to sustain the population and produce the next generation of soldiers or post-war citizens, the women are more important. Only women can bear children, and once they have done so they nurse them, and traditionally they raise them. On the flip side, women can get pregnant during deployment, which is rather inconvenient to the army. This isn't accusing women of having poor self-control: it takes two to tango, plus in a war zone sexual assault is common.
The last is another reason nations keep women off the front lines. Part of what a nation is defending in wartime is their people, and their belief that they are better than the enemy. They don't want their women bearing the enemy's children. They want them home bearing their own children. The psychological effect of the war on the population at home is very important. Israel has a universal draft, and initially found that women performed well on the front lines, but when they saw the terrible effect on national morale of women coming home in body bags, they pulled the female draftees back to support positions. It may have been something people would have become used to, as people are quite used to women delivering mail and performing surgery these days, but it wasn't something the country was willing to try. There's also the risk of the enemy catching onto this and using the female soldiers as hostages, taking advantage of a weakness men have, that urges them to protect women even when it's not the smartest thing to do.
Warfare, it seems, is such a full-being activity that it really does matter what sex-specific body parts you have. I'd suggest a universal draft, with deferrals available for the essential functions currently exempted, plus pregnancy, nursing, and other childcare responsibilities. Parents of weaned children could choose which of the couple would take the deferral. This would have the weird side effect of inverting the population growth patterns of previous wars: I'm sure plenty of women would get pregnant in order to avoid going to war. And then that would be perceived as unfair by men who had no such option. I guess there's no fair answer.