WASHINGTON—The Supreme Court on Monday turned away an appeal that sought to make it easier to erase student loans in bankruptcy, sidestepping an issue that has become a focal point for consumer advocates and lawmakers as millions of borrowers fall behind on their payments.

The court, in a brief written order, said it wouldn’t consider an appeal by an unemployed Wisconsin man who owes more than $260,000 in student-loan debt from business and law school. Mark Tetzlaff, 57 years old, said in court papers his alcoholism, depression...