Moxon (left) and Gale (right) were both criticised by Azeem Rafiq when the former Yorkshire player spoke to MPsDirector of cricket Martyn Moxon, head coach Andrew Gale and all members of the coaching staff are among 16 people to have left Yorkshire amid the racism scandal.
A report found former player Azeem Rafiq was a victim of "racial harassment and bullying" at Yorkshire.
Moxon had been Yorkshire's director of cricket since 2007.
Former captain Gale had been suspended as part of an investigation into a tweet he sent in 2010.
"Significant change is required at Yorkshire County Cricket Club and we are committed to taking whatever action is necessary to regain trust," said Lord Patel, the club's new chair.
"The decisions announced today were difficult to make, but are in the best interests of the club.
"Without making important changes to how we are run, we cannot move on from the past to become a culture which is progressive and inclusive."
Moxon, a former Yorkshire and England batter, was due to appear before MPs at the Digital, Culture, Media and Sport (DCMS) select committee on 16 November but was absent from work because of a "stress-related illness".
Rafiq told the committee that racist language was "constantly" used during his time at Yorkshire.
At the hearing, Rafiq said that the day after his son was still-born in 2017, Moxon "ripped shreds" off him in a manner he had never seen Moxon use when addressing anyone else at the club.
Rafiq detailed his experiences at Yorkshire, which left him close to taking his own life, for the first time in September 2020. After the findings of the report were released a year later, Yorkshire originally said no-one at the club would be disciplined.
Roger Hutton, Yorkshire's former chairman who resigned last month, said Moxon and former chief executive Mark Arthur "failed to accept the gravity of the situation". Arthur also resigned last month.
A Yorkshire statement said a new director of cricket will be appointed "imminently".
Gale made 156 first-class appearances for the club before giving up the club captaincy and ending his playing career at the age of 32 in 2016 in order to take up the coaching role.
Speaking at the DCMS committee hearing, Rafiq said Yorkshire became "toxic" after Gary Ballance took over as captain, shortly after Gale replaced Jason Gillespie as head coach.
He said Gale's attitude towards the death of Rafiq's son was that Rafiq was "making it more than what it is", and that "hardly anyone" asked after his and his wife's wellbeing.
Gale was suspended on 9 November after the Jewish News reported he sent a now-deleted tweet containing an anti-Semitic slur to former Leeds United head of media Paul Dews.
"We want to make Yorkshire County Cricket Club a place for everyone, from all backgrounds," Patel said.
"To do this, we need to rebuild our culture and instil positive values in everyone associated with Yorkshire. We are determined to learn from the mistakes of the past to become a club which people can trust.
