Milk is not as essential to our diet as we would like to think. My children rarely drink milk after about 15-18 months... I offer it to them and some days they drink it and some days not... Yogurt is a good, they even mix there cereal in yogurt rather then milk somedays.
If you are worried about him getting enough liquid, just keep offering drinks ever little while, little kids need liquid, but not as much as adults... Just make sure he is staying hydrated.
If you are worried about vitamins, you might talk to your pediatrician about a kids multivitamin (calcium is still essential but can be supplemented through other foods. But milk really is only a good source of vitamin D (The sun provides this) and Calcium witch can be gotten through other dairy products and foods.
Good luck, keep in touch with your doctor if you are worried and just offer drinks every little while, if he takes a sip of water ever little bit, he will stay hydrated and healthy and hopefully the phase will pass. If you start to notice constipation, I would go directly back to the doctor though. Because this could also indicate lack of hydration.
My son who didn't like to drink loved popsicles and so I would make my own and found he like plain water popsicles as much as any- It got water in him.... and even though I thought he was a little crazy for wanting to eat Ice, he loves it... He also refuses milk with his cereal, for this I offer the yogurt...
Good Luck, Your doing fine and your son will pull through the phase eventually... Just keep offering