I am a health care staff and on our unit everything is computerised. So when doctors prescribe it goes up on the program. However, there was this prescription that suddenly stopped on the program. So I asked the doctor if they had discontinued the medication as the medication had “come off” the program. They understood what I meant though, in fact I’ve heard others use “fall off”. Is “come off” alright to use in this case?
In the context you describe 'fall off' and 'come off' have the same meaning.
Their connotations are not quite identical, however. Consider the example of someone on a mailing list. If they do not respond to mail shots from that source, it is quite possible that their name will be removed from the mailing list. That is, they 'fall off' the list, not so much as against their will but regardless of it.
But if they had subscribed to the mailing list for x months, then, at the end of that period, they could be said to have 'come off' the list, in that case, as they had presumably planned to do.