Gale, 30, was banned for two games after Yorkshire beat Lancashire in the County Championship on 3 September and must now face a disciplinary panel.
The Daily Telegraph reports Gale allegedly used the word 'Kolpak' to refer to South African Ashwell Prince.
"We'll help him in any way possible," Moxon told the Yorkshire Post.
"We are going to do what we can to help him clear his name."
Gale was banned for his team's final two County Championship matches of the season under mandatory disciplinary procedure.
The England and Wales Cricket Board (ECB) has yet to confirm the nature of the hearing that is set to take place in the coming days, and has so far refused to comment on the disciplinary proceedings.
It is likely that the case against Gale will centre on whether his use of the word 'Kolpak' can be construed as being racist.
In Gale's absence, the White Rose county beat Nottinghamshire last Friday to win their 31st outright County Championship title.
Gale was informed on Thursday that he would not be allowed to lift the trophy with the team, but took to the field later on to join his team-mates.
Meanwhile, the county have confirmed that chairman Colin Graves, who is also deputy chairman of the ECB, will play no part in the disciplinary proceedings.
Gale's case will be handled by vice-chairman Robin Smith and chief executive Mark Arthur.
Yorkshire director of cricket Martyn Moxon said the county will support captain Andrew Gale's case against a charge of a racism offence.