How old is your baby? We had the same reaction to Baby Magic! But I know it was because my son is allergic to dairy. Right on the bottle it is called a Milk Bath and there is a milk protein in the ingredients. We stopped using it.
You may not know if your son has allergies yet if he is still breastfed and not eating food. Diary is a common one. I hear most kids outgrow it by age 3. We didn't know until 9mos when my friend was feeding her son yogurt and also fed my son. He broke out in a rash like hives all over his chin and around his mouth. After the bath, the same rash was on his back and stomach. It happened again another time with 1 bite and we knew. I was already not a big dairy eater/drinker so I just cut all dairy out of my diet (still nursing) and made sure I read all packages. Nonfat dry milk is in so many things! You get used to looking for it. He drinks soy or rice milk.
You could have him tested, or you could wait until he is eating solid foods to see what happens when he gets dairy. If he is allergic to dairy, plan to breastfeed longer to make sure he gets what he needs from your milk supply. My son is now 4 and starting to be able to tolerate some things that have less than 2% of a milk protein ingredient (it is in almost everything, btw!). He also knows he is allergic to dairy and will not eat anything he thinks has dairy in it! He even knows milk chocolate has milk in it and will give him a tummy ache.
You could take the body wash back, or give it to a friend!