Speaking to reporters on Friday, Francona noted that he's "proud" of the team for making the move.
"What we're really proud of is the first name of our team, which is Cleveland," he said. "Regardless of how we felt about it, what was important was how other people who it was affecting felt about it. I don't think anybody was ever trying to be disrespectful. But that wasn't a good enough answer anymore."
Cleveland owner Paul Dolan announced on Monday that the team would adopt a new nickname, but the change won't take place until after the 2021 season.
"Our role is to unite the community," Dolan told MLB.com's Anthony Castrovince. "There is a credible number of people in this community who are upset by our name, are hurt by our name, and there is no reason for our franchise to bear a name that is divisive."
Cleveland previously stopped using the Chief Wahoo logo on its jerseys and on signs in Progressive Field in 2019, though merchandise featuring the controversial mascot remains on sale in the stadium's team shop.
Dolan has yet to announce what the team's new nickname will be. He told Tom Withers of the Associated Press the organization is "going to take the time we need to do it right."
The franchise has used the nickname since 1915.