In an email that was sent later to company employees, Bonderman said, "I want to apologise to my fellow board member for a disrespectful comment," calling it "inappropriate".
Bonderman, the 74-year-old co-founder of a private equity firm, also apologised personally to Huffington.
"David has apologised to all Uber employees for a remark that was totally inappropriate and against the new culture we are building at Uber," Huffington said in a statement.
A few hours later, Bonderman resigned and released this statement: "Uber is examining the issues with its culture, and making significant changes and working to right what has been done wrong, which is extremely important for the future of the company. I do not want my comments to create distraction as Uber works to build a culture of which we can be proud. I need to hold myself to the same standards that we're asking Uber to adopt."
Huffington also issued a statement praising Bonderman for his decision, "I appreciate David doing the right thing for Uber at this time of critical cultural changes at the company."
The incident came as Uber sought to move past a leadership crisis that has led to the departure this week of chief executive Travis Kalanick for a leave of absence and the exit of his close ally and confidant Emil Michael, a senior vice-president at Uber.