Which Supreme Court case declared racial quotas could not be the only criterion in affirmative action plans?

Answer: Regents of the Univeristy of California v. Bakke (1978).

Regents of the University of California v. Bakke was a decision by the Supreme Court of the United States. It upheld that affirmative action allows race to be one of several factors in college admission policy. However, the court ruled that specific racial quotas were impermissible.

Although the case was highly publicized, most affirmative action programs in universities continued without any change.