Translation:I can't drink milk, but I can drink soy milk.
I agree. Most English speakers would say "cow's milk" if they're contrasting with "soymilk". Also, true milk doesn't hav to be cow's milk; it could be goat milk or sheep milk. And in the Chinese 牛奶, the part 牛 literally means cow, so it is implied that it is cow's milk.
豆奶 often has cow's milk in it as well, where as 豆浆 is generally just soy beans, water, and perhaps something to sweeten and flavor it.
"I can't drink cow milk, but I can drink soy milk" should also be correct.
In English we don't say "cow milk". We say "cow's milk". Other English speakers might say things like "dairy milk" or "real milk", but "cow milk" sounds a little childlike.