Are you on the Wonder? I believe the "fridges" in the rooms are mostly just cooler boxes, although some may have been upgraded. We had a non-upgraded one on the Magic recently or maybe it was just broken. It didn't keep things near cold enough to store milk. But you can ask your stateroom host to bring an ice bucket to the room twice a day, that will keep it well-chilled for a few hours.
Milk (skim, whole, chocolate, etc.) is also one of the items (along with coffee, croissants, toast, etc.) that you can pre-order to be delivered in the morning with the hang-tag that you find in the desk, fill out and leave on your door at night. This service is available every morning except the last morning. It's free other than gratuity.
I thought you could also order milk with your room service order (available 24 hours a day). However I don't see it on the menu so maybe I am mis-remembering that.