Is eivah - animosity - a good rationale to permit the violation of all issurei d'rabonon, or only when the gemara stipulates on a specific issur that it is permitted for eiva, would we allow it? The gemara understands that the issur d'rabonon of assisting a goy in child birth would be permitted for the sake of eivah. However, on shabbos it would not be permitted since the eivah can be avoided with an excuse that we only violate shabbos for those who keep it. The gemara seems to imply that had we not had a good excuse we would even permit the violation of shabbos to avoid eivah - animosity. Tosafos asks that we never find that an issur d'oraysa would be permitted for the sake of preventing animosity? Tosafos answers that we are speaking when the baby is almost out, so the pulling it out wouldn't constitute an issur d'oryasa, and we would only permit actions that would be d'rabonon. There may be an implication in Tosafos that we would permit all issurei d'rabonon to prevent eivah. However, the Ritva says that we can't permit all issurei d'rabonon for the sake of eivah unless chazal stipulated in the original gezeira that it should be mutar to avoid animosity.
However, it seems clear that even Tosafos holds we cannot freely be matir all issurei d'rabonon for the sake of eivah. Tosafos 2a discusses why we are lenient to allow business with goyim within 3 days of their holiday. Initially, Tosafos suggests that to avoid business with them would create animosity, which would allow the violation of an issur d'rabonon. But, Tosafos doesn't say it matter of factly, rather proves from a gemara 7b that this specific issur d'rabonon that prohibits business within 3 days of their holiday is permitted to avoid eivah. Tosafos continues by rejecting the proof - the story of R. Yehuda Nesiah doesn't indicate that it would be permitted to violate to avoid eivah, rather he was looking for a way to avoid both the violation while still avoiding the eivah. Now, if Tosafos would hold that eivah is a blanket heter to allow all issurei d'rabonon, the rejection of the proof shouldn't impact the din that for eivah it is permitted to do work with them. Clearly, Tosafos understands that without a proof from the gemara that this specific issur would be permitted to avoid animosity, we would have to assume that the concern of eivah isn't even strong enough to push off an issur d'rabonon (unless specifically stipulated by the gemara).
Similarly, Tosafos on the bottom of 26b quotes a gemara to prove that one is allowed to heal a goy if there is payment since otherwise it would create eivah. If every d'rabonon is mutar to prevent eivah, why does Tosafos need a proof from a gemara in gittin 70a that there are circumstances in which you can heal a goy? It must be that not every d'rabonon is mutar to prevent eivah.
Therefore, when R. Moshe (Igros Moshe O.C. 4:79) is matir nowadays chilul shabbos for the pikuach nefesh of goyim, he explains that eivah is not sufficient to permit it. R. Moshe permits it because the animosity will be so strong against Jews and will be so publicized in the media that it will lead to pikuach nefesh of a Jew - כפי המצב במדינותינו בזמן הזה איכא מצד איבה סכנה גדולה אף במדינות שהרשות לכל אדם מישראל להתנהג בדיני התורה וכו' ופלא על הגאון החפץ חיים (מ"ב ס' ש"ל ס"ק ח') שכתב דהא הרופאים אפילו היותר כשרים נוסעים כמה פרסאות לרפאות עכו"ם ושוחקין סממנים בעצמן, ומסיק דמחללי שבת גמורים הם במזיד אף שיהיה איבה מזה, הא ברוסלאנד בעיירות הקטנות שהיה רק רופא אחד לכל הסביבה הרי ודאי ברור אם לא היה הולך לרפאות את העכו"ם היה ברור שהיו הורגין אותו בטענתם שגרם מיתה לבנם ובתם וכו' ועיין בחת"ס חיו"ד ס' קל"א שכתב בפירוש ואם יש באיבה זו חשש סכנת נפשות יש להתיר אפילו מלאכה דאורייתא וכו' לכן פשוט שבזמננו יש לדון זה כסכנה ממש ויש להתיר כשנזדמן זה. This also enters into a realm of what qualifies as pikuach nefesh - since the individual who would be in sakana if we wouldn't save the goy cannot be identified (big discussion between the chazon ish y.d. 208:7 and nodeh b'yehuda regarding the hagdara of pikuach nefesh when there is no individual lying in front of you in need of being saved), it is more of a meta concern of sakana - וצריך עיון