That's exactly what Lee said about Lincoln in the mid-1800's.
Your analogy would hold if the CSA had claimed to be the legitimate government of the USA. However, the CSA never did any such thing and it acknowledged, at all times, that the government of the USA at the time was the legitimate government of the non-CSA States.
Going back to your President example, the Pope claiming to be the head of Orthodoxy would be like GWB claiming to be the Prime Minister of Australia.