A quick look around at some of the Rishonim seems to confirm the psak of the MG”A that women are not obligated in machatzis hashekel. Although it is not an aseh she’hazman gerama, there are other reasons to exempt women, among them: the mitzvah is given to “kol ha’over al hapekudim” which does not include women; the mitzvah is directed to “bnei Yisrael” to the exclusion of “bnos yisrael”; the mitzvah is to give “ish kofer nafsho” to the exclusion of “isha”. However, the language of the Mishna in Shekalim seems to suggest that there is a chiyuv. The Mishna (1:3) writes that we do not take a mashkon (collateral) from women for the machatzis hashekel, implying only that no coercion is used to get them to pay, but they are still obligated to donate. The She’arei Korban in the Yerushalmi takes issue with the MG”A based on this diyuk, but does not explain what he does with the Rishonim.
I noticed an interesting Alshich quoted in the footnotes to the Halichos Beisa to explain why a half-shekel and not a whole shekel is donated. The purpose of shekalim is a kaparah for the cheit haeigel, “ish kofer nafsho”. Since the women did not sin with the eigel, they are exempt from the need for kofer. Ishto k’gufo means a husband and wife count as one unit and would be required to donate one full shekel. Since only half of that unit sinned, the kofer amounts to only half a shekel.