But Ducey stopped short of calling Moore to resign, saying the final decision should rest with voters.
The Hill was one of the first to report Ducey's comments, which were made at a Phoenix food bank.
The accusations against Moore, first reported by an explosive story in the Washington Post, also apparently led on Wednesday to the resignation of Moore's communications director and spokesman, John Rogers.
However, even as Moore helps fuel a new cultural battle that arose in October with revelations against Hollywood mogul Harvey Weinstein, it's unclear whether Alabama voters are prepared to reject Moore.
Fivethirtyeight.com reported on Wednesday that an average of surveys on Moore's race with Democrat Doug Jones shows a statistical dead heat.
Ducey's reticence in calling for Moore's resignation puts him in the same category as President Trump, who said he was "troubled" by the allegations against Moore, but did not ask him to resign.
Ducey has also declined to push for the resignation of Arizona State Representative Don Shooter, R-Yuma. Shooter has been under fire by the accusations of multiple women, starting with Rep. Michelle Ugenti-Rita, R-Scottsdale, who claim he sexually harassed them.