You can only adopt or own a dog if you are a LEGAL adult; age 18 or OLDER. They cannot put the dog down before reading THE ADOPTION CONTRACT they signed. I will bet they have to RETURN the dog, if they no longer want it or find it suitable.
No, you DO NOT put a dog down for fighting WITH another dog IN the same HOME. That is an EXTREME overreaction. Instead, if you are responsible adults, you institute MANAGEMENT, and PREVENTION (now that you know this can happen) and obviously there is not ENOUGH obedience training - since neither dog could be called OFF. So everybody needs MORE training or a refresher course. And one or more dogs may be resource guarding, so speak to an animal behaviorist as to WHY this happened, which your original adoption center can help you with.
I would POINT OUT, that according to you - that the smaller dog INSTIGATED the attack. Again, (legally) if these dogs had been on leashes & owned by two different people -the larger more powerful dog, CANNOT be faulted for fighting back, when it DID NOT START the fight. That is INSANE! It is an "over reaction" to cover up GUILT or misplaced anger, to a lack of watchfulness or lack of responsibility - in allowing THIS TO HAPPEN. The humans in the home are 100% RESPONSIBLE, really not either dog. If they had left a loaded gun out or easily accessible, and a child had been shot while playing with it, would they then insist that the owner of the gun be put down? (I bet NOT!)
You can muzzle both dogs, and/or use a crate or indoor/outdoor kennel system for keeping the dogs APART (100% of the time, IMO) or being unable to fight WITH each other, if loose. Both dogs should be muzzled if loose at the same time, but I would rotate dogs, as to WHO is loose and divide up the house via: locked doors, Dutch doors, or TALLER, stronger baby gates or child doorway barriers (that you make sure Ice can't get over or jump).
PS having TWO male dogs in the same home, is never advisable, versus opposite gender dogs. I hope BOTH are neutered. If not neutering should be done, ASAP. Un-neutered dogs are 75% more likely to get into dog fights. It also is NOT recommended that breeds NOT BE more than 25 pounds difference, in size as adults, so as to prevent or lessen the chance of even unintended injuries, when playing. Your adoption group did not follow EITHER widely recommended suggestion, if they "KNEW" you had another dog - when you got Ice from them.