Trustees also warned President Gordon Gee that any future inappropriate comments could result in punishment including dismissal.
"The board will have no choice but to take such action," according to a copy of the March 11 letter obtained Friday by The Associated Press.
An AP report this week revealed remarks Gee made to the university's Athletic Council in December, saying that Notre Dame wasn't allowed to join the Big Ten because its leaders were not good partners. He criticized its Roman Catholic priests as "holy hell" on days other than Sunday and joked "those damn Catholics" can't be trusted.
He also questioned the academic integrity of schools in the SEC and the University of Kentucky.
Gee apologized in a statement Thursday and again on Twitter Thursday night.
"I am truly sorry for my comments_such attempts at humor do not reflect Ohio State values, nor my role as its president," the tweet said.
In their letter, the trustees laid out steps Gee must take including issuing personal apologies, getting help from professionals to revisit his personal communications and speechwriting processes and rethinking what speaking engagements he accepts.
Trustees told Gee that his attempts "to bring a bit of levity" to significant issues have had the opposite effect at times.
"As a result, instead of your words promoting and uniting us, they have sometimes embarrassed and divided us," trustee chairman Robert Schottenstein and trustee Alex Shumate, who led the search committee that hired Gee in 2007, said in the three-page letter obtained through a records request.
"Such comments are not befitting a great university like Ohio State or its leadership," the letter continued. It added: "Although we do not believe that you intended harm, such comments risk diminishing the effectiveness of our collective efforts and of your good work."
The letter said Gee is making progress on the board's list of requirements.
"Your willingness to seek guidance and counsel on multiple levels, from a variety of sources, on how to adapt and grow is a hallmark of your leadership style and one that we value highly," the letter said.
University and athletic conference officials have almost universally called Gee's remarks inappropriate but also said his apology has been accepted.
Associated Press writers John Seewer in Toledo, Tom Coyne in South Bend, Ind., Janet Cappiello in Louisville, Ky., Stephen Hawkins in Irving, Texas, and Mark Long in Destin, Fla., contributed to this report. Andrew Welsh-Huggins can be reached on Twitter at https://twitter.com/awhcolumbus .