Oh, I'm so sorry you are having to see your little guy not feeling well.
I am sure the reason they don't want him to eat is because he may get sick. I think you are doing the right thing by just giving him tiny amounts and the ice. I know ice is what pregnant women get in the hospital when they are delivering, so I'm sure it is o.k. for your little patient to have some as well!
Hang in there....tomorrow will be a much better day.
(I dread the day I will have to have my Sophia spayed.
)