The only time an autopsy may be performed is when it will somehow save a life. (Most, if not all, autopsies are not done to save lives.)
The reason I find this specific paragraph intriguing is because the law in the Shulchan Orech (Yoreh Deah 349:1) which all of this is derived from equates a Jewish corpse and a non-Jewish corpse. Why then are there so many special rules by a Jewish corpse, but a non-Jewish corpse can so easily be used?
I have not looked through all of the sources since I am limited with my referencing materials. However, using Hebrewbooks.org I was able to find a good Shulchan Orech, Yoreh Deah that had many commentaries. There, I was able to find the Pischei Teshuva on this halacha (Yoreh Deah 349:1) which gives a foundation to the seemingly lenient opinion by non-Jews, but the much stricter opinion by Jews. It says that the prohibition that one may not benefit from a Jewish corpse is biblical, but not benefiting from a non-Jewish corpse is only rabbinically prohibited.
This actually helps explain HOW one could be allowed to perform an autopsy on a non-Jewish corpse when there is minimal reason and a Jewish corpse would need a maximal amount of reason, but I still need to understand WHY it should be that one is biblically prohibited and one is only rabbinically prohibited. I hope to translate one or both of Rav Moshe Feinstein's Responsa on this subject in the coming days.