Would a world governed by women be more peaceful?

It's just as arbitrary as you thinking that a woman leader would be good because they are nurturing. Good soul doesn't mean automatically good choices.
So either you take a stern woman leader as an example of woman in charge, or you take the everyday woman (good luck finding out what does it mean :P) who's not in charge and try to find out how they would act in a similarly high position. Both of the options are useless, it's so far from the reality.

Hillary Clinton is a war hawk, and women generally have to prove they are just as "tough" as the men so expect that to override any instincts.

Its all opinions on a hypothetical scenario anyway. Leading in and of itself is a position of power over the ones that arent leading. I still say the power hungry chicks would end up being more ambitious in pursuing it than the rest.

For what it's worth, studies I've seen showed that instinctively, women attempt to resolve things through social means, and men through physical means.
However, there was no actual indication of differences in temperament- that women were more compassionate/et al, outside of social influences that shaped them to be such [and men to be less so].

Given that anything more concrete than that is pure speculation (given the pervasiveness of social influences), you may want to avoid the whole 'women are better than men' bit, even if you're attempting to make a 'but at this thing, rather than another thing' type of judgement- those kind of stereotypes hurt both men and women, who can easily be aggressive or compassionate in turn.

For what it's worth, my personal experience vastly contradicts your statement, as the majority of the women I've met have been more ruthless, aggressive, ill-tempered, and selfish. The majority of men were more calm, compassionate, nurturing, careful-minded, and steady. Neither group really exemplified any set of features to extremes, compared to the other.

Basically, the whole 'women are X', 'men are Y' thing is pure BS, since there's no evidence either way, nor even any overall indication of such in studies, even with social influences in play.

It's going to depend on the individuals and the environments, same as anything else.

Perpetuating that without any basis actually DOES do harm- it continues to convince women that they should resist their more determined natures, and it convinces men to favor more assertive, less-tender outlooks.
Meanwhile, perpetuating the 'women are more compassionate' mantra doesn't actually do women any good- even if it is true, it continues to emphasize gender roles, which prevent the progression toward equal assessment and the consideration of such traits- even if they ARE overpronounced gender-wide- of being more characteristic of the individual, than of gender-related social necessity.

The first step should always be "that person is compassionate, that person is an aggressive dick," not "women are, men are, blacks are, whites are".
I mean, there are clear genetic factors that factor certain races, but it'd be pretty ridiculous to call those out, right? So why is it okay to call out sexual stereotypes, when they, unlike commonly pronounced racial traits, don't actually have any scientifically verifiable basis?

While there are tons of minor influences on men and women on the whole, dominantly studies I've encountered and simple observation of existing counter-examples show that social influence- or rather, the inclination to cater to expected social appearances- dominates the creation of any major deviations in behavior.

http://www.nytimes.com/1994/03/29/science/sexes-equal-on-south-sea-isle.html?pagewanted=all
http://link.springer.com/article/10.1007%2FBF01420990
Actual egalitarian and matriarchal cultures- even though they do still have social expectations on what behaviors are suitable to gender roles- suggest that behaviors in an equalized environment are, in fact, pretty indistinguishable. In fact, in the limited data presented, women are shown to be more inherently aggressive than men- something that other studies have ALSO suggested.

tl;dr version:

1. The science does NOT support any firm indication that there any notable inherent temperament differences between men and women, and may even support women being more aggressive than men by nature.
2. The science supports the main influence on an individual's behavior being social influences, rather than genetic factors relating to gender.
3. All signs indicate that perpetuating gender-specific stereotypes has a negative influence
4. Your expectations on gender actions are based within your own social experience, and thus flawed in overall perception, especially given #2.
5. Likewise, my own experiences contradict yours fully, adding further support to the non-absolute application of the assertion.

So yeah.. you may wanna reconsider your viewpoint some :P

As far as the original question goes..
Well, I'll paraphrase it a bit, since the original question is completely impossible to answer :P
"Would women do better in charge than men?"
The answer to that is simple: As soon as we're able to stop asking absurd over-generalized questions like that, we won't have to worry about those questions anymore, because they'll be irrelevant.

Finally, and interestingly-
None of this actually needs to make you sway from your opinion [not, apparently, that anything could make you do so].
After all, we DON'T live in an egalitarian society, and that fact may make in fact make women in our current society present more factors which are positive for the role of leadership than men.

That's all supposition, of course, but if that's your opinion- well, like with most gender-based things, there's no real evidence either way :P
So by all means, believe that, support that, promote that- and in a society where women are treated notably inferior, supporting women for leadership certainly can't be said to be a bad thing.

Just keep in mind that you're not necessarily working with absolutes, however. :)