Gemara

Beraisa: We do not Mechalel Shabbos for a Choleh through Nochrim or children (this is the correct text), rather, through adult Yisraelim. We do not do Piku'ach Nefesh via women or Kusim, but (some texts - because) they join other opinions (this will be explained).

Shabbos 128b - Beraisa: We Mechalel Shabbos for a woman about to give birth...We bring oil for her if she needs. If it suffices to bring oil on another woman's hand or in her hair, we do so. If not, we may bring it in a Keli.

133a - Mishnah: One may do all needs of Milah on Shabbos...If one did not grind the cumin before Shabbos, one chews it with his teeth and puts it on.

Rishonim

Rif: We do not Mechalel Shabbos for a Choleh through Nochrim or children, rather, through adult Yisraelim. We do not do Piku'ach Nefesh via women or Kusim, because they subscribe to other opinions.

Tosfos (DH Ela): When possible we do not use Nochrim lest they neglect to do what is necessary, further endangering the Choleh.

Ran (DH v'Ein): We do not Mechalel through Nochrim or children at all, even on one occasion, lest onlookers think that Benei Mitzvos may not Mechalel Shabbos. In another case when Nochrim or children are not available, these people will be reluctant to Mechalel Shabbos. Even though needs of a Choleh who is not dangerously sick are done through Nochrim, we do not use Nochrim for Piku'ach Nefesh. Even though women and Kusim are Benei Mitzvos, we do not say that specifically they should Mechalel Shabbos, for women and Kusim are prone to think that they are being used because it is really forbidden.

Other texts say that we do not use (specifically) women or Kusim, but they join other opinions. I.e., we do not entrust Piku'ach Nefesh solely to women or Kusim lest they be lazy. They may participate with male Yisraelim.

Rashi explains that we do not Mechalel Shabbos based on the opinion of women or Kusim, but they join other opinions. I.e. if a minority of male Yisraelim think that Chilul Shabbos is necessary, we include women and Kusim and Mechalel Shabbos if at least half of all the opinions think that it is necessary.

Question: We find that women and midwives are trusted (by themselves) about Chilul Shabbos for a Yoledes!

Answer (Ran ibid.): This is an exception, for women are more knowledgeable about the needs of a Yoledes than men are.

Ran (ibid.): It is clear that if a woman is sure that Chilul Shabbos is necessary and everyone is unsure, we follow her opinion. (This is unlike Rashi.)

Rambam (Hilchos Shabbos 2:3): We do not Mechalel Shabbos (for a dangerously sick person) through Nochrim, children, slaves or women, lest Shabbos will be light in their eyes. Rather, adult Yisraelim and Chachamim are Mechalel. Shabbos is like a weekday regarding such a person.

Kesef Mishneh: The concern 'lest Shabbos will be light in their eyes' applies only to slaves and women. The Rambam says 'Chachamim' to exclude women.

Mishnah Berurah (328:34): When Chachamim are present it is better that they Mechalel Shabbos.

Rambam (ibid. 11): We Mechalel Shabbos for a woman about to give birth...We bring oil for her if she needs. If it is possible to carry it through a Shinuy, e.g. hanging from another's hair, we do so.

Magid Mishnah: The Ramban holds that for any dangerously sick person, when possible we do all Chilul Shabbos through a Shinuy. The Rambam says this only regarding a Yoledes, but regarding other Cholim ' Shabbos is like a weekday'. Presumably Chachamim were stringent regarding Yoledes because labor is natural. Not one out of 1000 women dies from it.

Rosh (8:14): We do not Mechalel through Nochrim or children, rather, through adult Yisraelim. Even if there are children available, we are concerned lest on another occasion people will delay to find children.

Some say that we do not Mechalel based on the opinion of women or Nochrim, but they join other opinions.

Rebuttal (Riva brought in Rosh, according to text of Bach): If women are knowledgeable about Yoledes or Piku'ach Nefesh, we rely on them just like we rely on men! Rather, the text should say 'we do not say that we may not Mechalel Shabbos based on the opinion of women or Nochrim, but they join other opinions', i.e. they may even contradict what male Yisraelim say.

Question (Rosh): According to the Riva, it should say 'they even join other opinions!'

Rosh (ibid., according to Beis Yosef): We do not use women or Kusim lest they think that we ask them because there is no Heter to Mechalel Shabbos, and they will refrain from doing so.

Rebuttal (Tif'eres Shmuel): If so, the Rosh should say that we are concerned lest Shabbos be light in their eyes, like the Rambam says! Rather, we are concerned lest they think that men do not care if women Mechalel Shabbos, and therefore the women will refrain.

Poskim

Shulchan Aruch (OC 328:12): When we are Mechalel Shabbos for a Choleh on Shabbos we strive not to do so through Nochrim, children or women, rather, through knowledgeable adult Yisraelim.

Beis Yosef (DH v'Chosav Rabeinu): R. Yerucham says that if women or Kusim themselves are Mechalel for the Choleh we do not protest. If there is no one else available we ask them. This is clear.

Kaf ha'Chayim (70,73): According to the Rambam, we are concerned lest Shabbos be light in women's eyes. Perhaps even if they themselves want to Mechalel we protest. We prefer knowledgeable Yisraelim who will not Mechalel on another occasion for a Choleh not in danger.

Rema: Some say that if without delay it is possible to do what is needed without Chilul (mid'Oraisa) through a Shinuy or through Nochrim; this is our custom. If we are concerned lest the Nochrim will be lazy we do not rely on them.

Source (regarding Shinuy) - Tosfos (Shabbos 133b DH Lo'es): Even though we may Mechalel Shabbos for the baby after the Milah, we use a Shinuy (to grind the cumin with one's teeth) when possible.

Mishnah Berurah (36): If a small delay poses no additional danger to the Choleh, we use a Shinuy even if it entails a small delay, for then the Chilul is only mid'Rabanan. The Rema allows using children just like he allows Nochrim.

Gra (DH v'Im): The Beraisa forbids Nochrim only when there is concern lest they will be lazy (but not if a Yisrael supervises - Kaf ha'Chayim 76).

Rebuttal (of Rema - Taz 5): If we use Nochrim, perhaps next time people will delay until they can find a Nochri!