No.

The Romans were in many respects a bigoted and prejudiced people. They detested barbarians, thinking them somewhere on the scale of being above animal and less than human. They had no compunction ridiculing barbarian customs and believing that they were incapable on certain things.

However, the Romans did not have a concept of skin-color based race derived from biological similarity. That is a concept developed in early modern Europe, in the context of the encounter with the "New World" and the peoples who had been living there (and also, per a new and exciting book, the Spanish fascination with horse and dog breeding). The Romans rather thought of ethnicity as a package that primarily comprised language, dress, and mutable physical characteristics like hairstyle. Barbarians can be seen as barbarians because they wear barbarian clothes and they speak a non Greek / Latin language. (For some specific examples of this see, off the top of my head, the discussion of Vandals in Carthage in Jonathan Conant's "Staying Roman"). People who had an undeniable "racial" (in our view) heritage from a non-Roman group often rose to positions of high power within the empire. What they had to do was learn Latin, ideally also some Greek, dress like a Roman, say they were a Roman, and demonstrate talent. While as far as I know there were no sub-Saharan African emperors, a major dynasty, the Severans, came from what is now broadly Tunisia, and they had dark complexions and tight, curly hair, which is faithfully represented on their coins.

Here's a link to Severan coin, it won't let me put a pic: https://i.ebayimg.com/images/g/Jj8AAOSwtGlZNXbN/s-l300.jpg

This may seem crazy to us, given how deeply biological / color based race is embedded in the way that we view the world, but the Romans just didn't really see the world that way. To my knowledge there is only one "racist joke" in Latin literature. It comes from Petronius' Satyrica, as our heroes are trying to hide on a ship. They need to disguise themselves, and one has (in Ruden's translation an idea:

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"Eumolpus is a scholar. He takes ink with him wherever he does. We can use this to change our coloring, from our hair clean down to our toenails. That way we can be Ethiopian slaves and parade about cheerfully, without the tortures of hiding. Our dusky complexion alone will deceive our enemies!"

"Huh!" said Giton. "Why don't we circumcise ourselves to look like Hebrews, or pierce our ears like Arabs? Smear some chalk on our faces, and any Gauls we meet will take us for their own. As if color alone can shift a person's looks beyond recognition! No, everything has to agree, the disguise has to look plausible from every angle. For argument's sake, let's say the dye lasts on our faces. Say that the sea spray doesn't splotch us and the ink doesn't stick to our clothes, even though sweat alone makes cloth cling to us. Given all that, can we make our lips swell in that ugly way? Can we frizz our hair with curling irons? Slice our foreheads so that we get scars? Splay our knees out sideways? Walk on our ankles? Produce those foreign beards? Tinging the skin does nothing to change the body. Listen, I'm too scared to do anything but this: we should wrap our cloaks around our heads and jump overboard"

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Again, I am not totally versed in the entire corpus, but here is the one sort of racist against black African comment that you get in Rome. And even here, you can see, with things like circumcision and earrings, most of the other ethnic markers are body modification or fashion. The Romans focused a LOT on those things (especially barbarian hairstyles, which get popular and cause stirs) but they didn't really categorize based on biological, color-based race the way that we do. Race as such is an invention of the long 16th century.

Thanks! Very informative!