In a daylong meeting broadcast online this month, David B. Ashley, the president of the University of Nevada at Las Vegas, was repeatedly grilled, and then publicly demoted, during a contentious gathering of the university system's Board of Regents. The dressing-down followed months of public disputes over the conduct of Mr. Ashley's wife and heated exchanges between him and the university system's former chancellor, James E. Rogers. Left in its wake were an ousted president with a tarnished reputation and a state that could have a tough time attracting new leaders. Nevada law requires personnel sessions regarding university presidents to be held publicly. But the policy could harm the state higher-education system as it looks for a replacement for Mr. Ashley. "It's going to be very hard under this combination of circumstances to get a rational person to agree to look at that job," says one leading higher-education consultant, who asked not to be identified to protect job prospects.