It depends entirely how how you supplement to improve the immune system. This is an auto-immune reaction - an unusual reaction to something that should be harmless. Most of these sensitivities can be eliminated or significantly reduced. If someone has an anaphylactic reaction, that's much more difficult to eradicate but at least the person can be brought to the point of not having a life-threatening reaction through accidental cross contamination. Usually dairy allergies are not anaphylactic.
Outgrowing on his own? Maybe, maybe not. A lot has to do with the reduction and elimination of the inflammation that's already there, and the overall absorption of all nutrients to put the body back in balance. You can wait it out, or you can do something proactive to reestablish the balance that should be there but isn't. That's not usually a medical approach.
The science of epigenetics has really addressed a great number of immune system problems. Many medical doctors dismissed it for years but now most are changing their tune. It's not their fault really - most have not studied nutrition or non-pharmaceutical approaches in medical school. But with advances in everything from cholesterol reduction to cancer prevention to work in diabetes and neuromuscular diseases (everything on the autoimmune continuum), there's a giant shift in the thinking. That's why a lot of parents aren't doing as much allergy testing or NG tubes and all that, but moving to a preventive nutritional approach.