Where do you find that he recognized his mistake? It is my understanding she withdrew, he didn't ask her to withdraw.
You don't think they talked it over before she withdrew? I won't venture to guess which of them broached the subject first, but I will venture to guess that they concurred that it was the best thing for all concerned.
They consulted, and she decided to withdraw. It would be speculative to wonder which of them raised the issue first. What matters is that she offered to withdraw, and Bush agreed.
I would venture to suggest that both of them could see the handwriting on the wall and understand that the nomination was a mistake.