As deputy commissioner, Mr Bush said Mr Hutton's service record spoke of the officer's integrity, adding: "It is a great tragedy and irony that a man of such character should have been subject to devastating accusations of dishonesty."
The comments were drafted at police headquarters for Mr Bush the day before he travelled to the funeral to deliver the eulogy.
The Herald-DigiPoll survey asked people which of a range of statements best matched their view. Of those, 53 per cent said either an apology by Mr Bush over the comments, or his career contribution, were enough to get past the eulogy comments.
But 30.9 per cent of people agreed with the statement it was "unacceptable and he should not get the job".
Mr Bush is coming into the job of commissioner as the police prepare to release the results of a three-and-a-half year review of the case - and a review of allegations of malpractice by Mr Hutton.
Mr Bush has promised to distance himself from the review, which was sparked by the Crewes' daughter, Rochelle, issuing a plea for answers to Prime Minister John Key.
Mr Thomas' brother Des, who complained about Mr Hutton's role in the murder inquiry, said Mr Bush had not apologised to those affected by the eulogy. "He just said he apologised if he offended anyone. We know who he offended - and he should apologise to Arthur."
Des Thomas said the eulogy had stunned the family.
"We were wild and distraught."
The comments had seriously dented whatever faith they might have in police reviewing the murder inquiry and handling complaints against Mr Hutton and other officers involved in the inquiry, he said.
But Police Association president Greg O'Connor said the 30.9 per cent result was quite promising. It showed Mr Bush had the support of the majority of the public and he was known to be popular among police.
"There are plenty of other ways he will be judged [as] commissioner."
A spokesman for Mr Bush said he "declines the opportunity to comment".
