I think to answer this question first the concept of "racism" needs to be defined. Romans could of course be prejudiced against people with darker skin, or Greeks against "barbarians," but they were not racist.

Race is a category of definition within a scientific discourse. So race was not "invented" until the Enlightenment, or the "scientific revolution," both of which, coincidentally (or not) focused their attention on race alongside the growing practice of taking peoples from Africa and enslaving them in the Atlantic slave trade.

During the Enlightenment many, many, many, thinkers became preoccupied with categorising the different "types" of people Europeans began to encounter during a period of colonial expansion. However, these classifications were then placed in a hierarchy. Most famously in the Enlightenment "great chain of being" placing northern Europeans on top and peoples from Africa on the bottom, sometimes even in the subhuman category (this of course nicely justified practices of slavery as peoples from Africa were seen as absolutely inferior. As well as justified imposing tyrannical rule in places like India ).

When science really hardened into its contemporary form with observation, experiment, etc in the 19th century racism also hardened. Here we get the fields of phrenology and physiognomy (where the way someone looked expressed their mental capacities and essential characteristics). It should be noted that this science also applied to lower and "criminal" classes of Europeans, and these sciences basically tried to find who had "reason" and who did not, who could participate in society in law, politics, etc, and who could not.

All of this science that examined "racial" differences was then used to justify discriminatory laws, social practices, etc. Thus, slowly, in European culture structures from the 17th century on began to change to support race theories. Here we can say that racism is a system that privileges one "race" over another supported by laws, politics, and science. This did not exist in Ancient Rome, which is why some people would say Romans were not racist, because racism, that is a society built around race, did not exist because race as a biologically determined category did not exist.

With respect to the bible, the bible was used heavily to justify the practice of racialized chattel slavery as "natural." It's not that Protestants had racist thoughts, it's that in a racist society (one that invented and legitimised racial differences) the bible was used to further support theories of racial difference.

To say that the "notion of racism is antithetical to much of western ideology" misses the mark a bit, as there is no such thing as a monolithic "western ideology" as there was certainly no "anti-racist" discourse that pre-dated racialized chattel slavery.

There was also no such thing as "racist philosophy," but racial difference was something that began to be taken for granted from the 17th century on. It would just be a category of classification like gender (which has a longer history in the west, but we will not get into that here). Race was seen as something that determined a specific outcome, so if you were from Africa, especially from the 18th-early 20th century (and unfortunately still now) you were considered to have certain, negative characteristics. Racial difference permeated all aspects of Western society, it was not a unified philosophical discourse.

So to answer your question it was really the Enlightenment and the "scientific revolution" that examined, reified, and legitimised racism- that is social structures that discriminated against peoples with a certain amount of melanin in their skin.