WASHINGTON—The Supreme Court on Wednesday seemed likely to curb racial preferences in university admissions, after 80 minutes of intense argument revealed deep fissures among justices' views on the pursuit of diversity in higher education.

The case arose when Abigail Fisher, an applicant to the University of Texas, alleged the school rejected her because she is white. But the implications stretch far beyond the Austin campus. Since the 1970s, racial and ethnic diversity has become...