With viral diarrhea, the most important thing is supportive care. You want to keep your child hydrated as they're losing alot of fluid, so you need to replace it. Again, it doesn't have to be all at once. You want to stay away from all dairy products because when you have diarrhea, it wears away the lining of the intestine that absorbs lactose, and lactose is the sugar in dairy products. If your child only wants milk, they can have soy milk or soy formula, which does not have lactose, or lacto free milk. But in general, it's probably best to stay with clear liquids like gatorade, watered down juice and electrolyte solutions like that when your child has diarrhea. If your child wants to eat, you want the food to stay very bland,giving them foods that bind such as really starchy foods. They call this the BRAT diet: bread, rice, applesauce, toast, pasta. But don't be surprised if your child doesn't want to eat. Very commonly with these stomach viruses, you're child is going to have a crampy belly pain, the sort of belly pain where they say, "it hurts all over", and in between they're running around. In those cases, it doesn't matter if they don't eat.
It doesn't matter if they go for about a week but I wouldn't, you want to keep them hydrated because that keeps them out of the hospital and away from IV fluids. With solids, if they want it, great but if they don't, don't push it.
I hope this has helped with the question you have answered about your what seems to me an intolerance to milk problem.