The case in which the Supreme Court decided that obscenity was not protected by the First Amendment was

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The Supreme Court's decision that obscenity was not protected by the First Amendment was made in the case of Roth v. United States in 1957. The Miller v. California case in  1973 established a three-part test still used by courts to determine whether speech is obscene, but it did not overturn the ruling in Roth v. United States. Eisenstadt v. Baird and Griswold v. Connecticut both dealt with issues related to reproductive rights and privacy, not obscenity.

The Case in Which the Supreme Court Decided That Obscenity