WASHINGTON—The Supreme Court revisits affirmative action in college admissions Wednesday, the latest racially focused case to go before a court that has pared back government programs intended to benefit minorities.

Two years ago, the court passed up a chance to undo affirmative action at the University of Texas, instead asking a lower court to decide whether the Austin campus adequately justified classifying individual applicants by race. Now the case is back at the high court.

Under Chief Justice John Roberts, the justices have, on issues such as campaign finance and voting rights, taken a partial step in an initial case that later paved the way for a more consequential opinion shifting the law to the right.

The court is hearing the appeal at a time of student unrest at schools such as the University of Missouri over racial issues. The nation’s public universities say the stakes are high.

The lawsuit was filed on behalf of a white applicant, Abigail Fisher, who was rejected by UT in 2008. Ms. Fisher’s lawyers, in her latest appeal, said the Supreme Court should strike down the school’s admissions policy, making clear the Constitution “does not permit the use of racial preferences in admissions decisions where, as here, they are neither narrowly tailored nor necessary to meet a compelling, otherwise unsatisfied, educational interest.”