With the antifreeze looking thick and milky, can this damage my motor?

If your anti-freeze is milky, you have a problem; possibly a blown or leaking head gasket. Take your car to a good mechanic for diagnosis. In the meantime, drive it as little as possible - you could do more serious damage to the engine.

It is more likely to damage the automatic transmission due to coolant entering the ruptured *transmission fluid cooler* inside of the radiator .. then mixing with the transmission fluid .. Glycol (coolant) WILL destroy the transmission!