Final answer:
The ruling that schools cannot prevent students from protesting the Vietnam War was Tinker v. Des Moines Independent Community School District, which upheld students' First Amendment rights on public school grounds.
Step-by-step explanation:
Supreme Court Ruling on Student Protest Rights
The Supreme Court ruling that held schools could not prevent students from protesting the Vietnam War is Tinker v. Des Moines Independent Community School District. In this landmark case, the Supreme Court recognized that wearing armbands as a form of protest on public school grounds is protected by the First Amendment. This decision affirmed that students do not lose their First Amendment rights to freedom of speech when they step onto school property.
The case originated when students planned to wear black armbands to support a truce in the Vietnam War, which led to their suspension from school due to a policy created by the school principals. The Supreme Court ruled that the armbands represented pure speech and could not be prohibited unless they materially and substantially interfered with the operation of the school, which was not the case here.