While milk can be nutritious, it isn't absolutely necessary for a healthy diet. Milk has protein and calcium and is fortified with vitamin D. Other dairy products, such as cheese and yogurt, can provide the same nutrients, as can "alterna-milks" such as soy milk or almond milk, although you should talk to pediatrician before you switch to one of those.
If your child doesn't like dairy products or alterna-milks, he or she can get protein from meats, fish, eggs, soy beans, and other beans. Fish with bones, such as salmon, sardines or herring, have calcium, as does broccoli.
Vitamin D is made naturally by the body in response to sun exposure (a healthy amount—check with your pediatrician) and supplements can also be helpful.
So—don't panic. With a bit of creativity, your child will be fine.
Additional Information from HealthyChildren.org: