Assisting a gentile women give birth:
Is forbidden even for reward as one can excuse herself without causing enmity: One may not assist a gentile to give birth even if the needed assistance does not involve desecrating Shabbos, such as [in a case that] the baby has already detached [from the womb and begun] to exit, in which case removing it does not involve [the prohibition of] uprooting an item from its area of growth. Furthermore, [even] if the Jewish woman is a known midwife and she receives payment [for assisting in birth], in which case it is permitted for her during weekdays to assist gentile women in giving birth due to [that refusing to do so will cause] enmity, nevertheless on Shabbos [since] there is no enmity [involved in refusing to assist, and thus it remains forbidden to help a gentile woman give birth].
The reason that there is no enmity on Shabbos is because: the Jewish midwife is able to excuse herself [from assisting without causing hatred] by saying “Our women which keep Shabbos are permitted to be assisted in giving birth on Shabbos, however your women which do not keep Shabbos it is forbidden for us to assist them in giving birth”.
The Law by an Arab woman: [Furthermore] even an Arab woman, which is not an idolater, (and there is thus no prohibition to assist them in giving birth during the week as there is by idolatress gentiles in which it was forbidden due to that she is assisting a idolatress child to be born, nevertheless on Shabbos) it is forbidden [to assist her] due to it being an unnecessary act, as will be explained in Chapter 332 [Halacha 1], (being that it is possible to excuse oneself from giving assistance without causing hatred).
The law by Karite women: However the Karaim since they keep Shabbos it is permitted to assist them in giving birth (in exchange for reward [and only] in matters that do not contain desecration of Shabbos. [This may be done] due to enmity)
If one is not able to excuse herself without causing enmity: Even by idolatress gentile women if it is impossible to excuse oneself [from assisting] without causing enmity, it is permitted [to assist] in matters that do not contain desecration of Shabbos.
Summary- Assisting a gentile women give birth:
It is forbidden to assist any gentile women to give birth on Shabbos in any way even if it does not involve a Shabbos desecration. However if one is unable to excuse herself from assisting without causing enmity, then it is allowed to assist her in matters that involve no Shabbos desecration.
Practically today may one assist a gentile woman in giving birth, and may he desecrate Shabbos on her behalf?
Yes, as due to advanced communication and the constant anticipation of reasons to use to hate and attack Jews the matter can lead to actual danger for other Jews if one were to refuse to treat them. It is thus allowed to even do Biblical transgressions. However some have written that while doing so one is to constantly think that he is doing so simply to save himself or other Jews from danger, and then it is considered like a Melacha Sheiyna Tzarich Legufa which many Poskim allow to help save a gentile.
 Even one who does not sever idols, such as a Muslim
 due to it being an unnecessary act, as well as if she is an idol worshiper it is forbidden even during the week due to that she is assisting a idolatress child to be born.
 Piskeiy Teshuvos 330/2
 So rules: Chasam Sofer  Yoreh Deah 131; Divrei Chayim 2/25; Igros Moshe 4/49; Chelkas Yaakov 2/54; Minchas Yitzchak 1/53, 3/20
 Shearim Hametzuyanim Behalacha 92/1; Titz Eliezer 8/15-6