The general reason for WHOLE milk is: that the "milk fats" in it is a crucial nutrient... for brain development.
"Fats" is important also for the nerve development as well.
It is not just for the calcium.
And thus, it is recommended for Toddlers through 2 years old.
At which time, this is a building block period, for the body's development etc.
But, your child has to be diagnosed conclusively, that milk is causing his Eczema.
There are many triggers, for eczema.
For example: My Mom gets eczema. BUT not from milk. She gets it from anything with Oats/Oatmeal in it. So if she avoids this... she does not break out.
Everyone, is different.
Eczema, can be triggered by topical or ingested things.
And maybe even his vitamins are causing the eczema? Because, some vitamins have paraben in it, which can cause allergies in some people. Parabens are used to preserve some things. Or it can be a constituent of those gel caps.
For example.
And, vitamin C as well, can create rashes in some people.
My Mom as well, if she eats oranges or "C" things, she gets Eczema or rashes. For example.
Here is a resource/site you should check out:
http://www.nationaleczema.org/
No matter what "milks" you give your child, you also need to make sure he is getting enough of the healthy "fats" in his diet. Because, it impacts brain development and nerve development etc.
Again, your Doctor did not conclusively say it is milk. The Doc is just guessing.
Also, be careful of giving him too much vitamins/quantities. Because, it can overload the kidneys. Especially in a baby or very young child.