WASHINGTON—Civil rights officials at the U.S. Education Department are requiring the Texas Tech University Health Sciences Center medical school to cease factoring race into admissions decisions, putting other institutions of higher education on notice that their continued use of affirmative action policies will draw federal scrutiny.

The mandate came in a deal the medical school struck with the Education Department in February, concluding a 14-year federal investigation into the university’s use of affirmative action, according...