Final answer:
The case Engel v. Vitale ruled that requiring students to recite an official school prayer was unconstitutional, playing a significant role in establishing the separation of church and state in public schools.
Step-by-step explanation:
The case that ruled it was illegal for states to require official school prayers to be recited by the students is Engel v. Vitale. In the landmark decision of Engel v. Vitale in 1962, the Supreme Court held that the practice of starting the school day with an official prayer, even if it is nondenominational and voluntary, was unconstitutional as it violated the Establishment Clause of the First Amendment. This decision established an important precedent in separation of church and state policies and prohibited organized prayer in public schools, setting a clear boundary between the business of government and the endorsement of religious practices within public institutions.