He also complained about Commissioner Peter Marshall, who he believed failed to intervene and discipline Bush.
A spokeswoman for the authority, Rebekah Palmer, said it was taking no action. "The authority considers that, while the appropriateness of the comments may be questioned, its view is that, under the circumstances, they do not amount to a matter of misconduct on the officer's part.
"The actions of Deputy Commissioner Bush would appear to fall within the category of a performance issue, something that falls outside the authority's mandate and remains the prerogative of police," she said. "The matter has been dealt with by Commissioner Marshall and is now closed. The authority does not see any requirement to investigate the commissioner himself."
Marshall told the Herald on Sunday: "The authority has indicated they are happy to leave the matter with me to deal with as I see fit. As Commissioner, I have already made my views clear on the matter.
"With the current police review of the Crewe file under way, it would be inappropriate for me to engage in any more public debate on the matter," he said.
The police review of the original Crewe investigation, being carried out by Detective Superintendent Andy Lovelock, is nearing completion.