I wouldn't say that there wasn't "racism" in Ancient Greece and Rome. This comes from Vitruvius' On Architecture (Book 6.3-4):

3. This, however, is determined by consideration of the nature of the place and observations made on the limbs and bodies of the inhabitants. For where the sun acts with moderate heat, it keeps the body at a temperate warmth, where it is hot from the proximity of the sun, all moisture is dried up: lastly, in cold countries which are distant from the south, the moisture is not drawn out by the heat, but the dewy air, insinuating its dampness into the system, increases the size of the body, and makes the voice more grave. This is the reason why the people of the north are so large in stature, so light in complexion, and have straight red hair, blue eyes, and are full of blood, for they are thus formed by the abundance of the moisture, and the coldness of their country.

4. Those who live near the equator, and are exactly under the sun's course, are, owing to its power, low in stature, of dark complexion, with curling hair, black eyes, weak legs, deficient in quantity of blood. And this deficiency of blood makes them timid when opposed in battle, but they bear excessive heat and fevers without fear, because their limbs are nourished by heat. Those, however, born in northern countries are timid and weak when attacked by fever, but from their sanguineous habit of body more courageous in battle.

So while there isn't exactly a focus on skin color, there is a focus of why certain people act a certain way in respect to their geography. Vitruvius argues that the Italians are best for the reason of being in between both extremes, from section eleven of the same book:

11. on this account the people of Italy excel in both qualities, strength of body and vigour of mind. For as the planet Jupiter moves through a temperate region between the fiery Mars and icy Saturn, so Italy enjoys a temperate and unequalled climate between the north on one side, and the south on the other. Hence it is, that by stratagem she is enabled to repress the attacks of the barbarians, and by her strength to overcome the subtilty of southern nations. Divine providence has so ordered it that the metropolis of the Roman people is placed in an excellent and temperate climate, whereby they have become the masters of the world.

I wouldn't argue that this is the standard definition of racism that we have now but it does have a small semblance of what we would think of as racism.