AUSTIN, Texas—Since Wallace Hall Jr. was appointed to serve on the board of the University of Texas in 2011, he has sought thousands of pages of records to try to root out potential wrongdoing at the university.

Now, the Dallas businessman faces possible impeachment from state lawmakers after a legislative probe concluded Mr. Hall went too far, improperly using confidential student records and filing unduly burdensome document requests. "While serving as a regent, Hall acted like a roving inspector general in search of...