Final answer:
The NY "Released Time" Program would violate the Establishment Clause of the First Amendment, which is supported by the 1962 Supreme Court decision Engel v. Vitale, prohibiting state-sponsored prayer in public schools and establishing a clear separation between government and religious activities in public institutions.
Step-by-step explanation:
The NY "Released Time" Program refers to the practice of allowing public school students to be released from school during school hours to participate in religious instruction or prayer. The question addresses whether this program violated the Establishment Clause of the First Amendment, which prohibits the government from making any law 'respecting an establishment of religion.'
The Supreme Court case Engel v. Vitale (1962) directly addressed the constitutionality of a state-sponsored prayer in public schools. The decision indicated that by providing the prayer, New York officially approved religion, which is unconstitutional according to the Establishment Clause. This established a critical precedent that prohibits organized prayer in public schools and has become a cornerstone in the separation of church and state doctrine.
Subsequent cases, such as Employment Division v. Smith and Santa Fe Independent School District v. Doe, upheld that neutral laws of general applicability and student-led prayers also must comply with the Establishment Clause. Therefore, based on the Engel v. Vitale decision and later interpretations, the NY "Released Time" Program would be seen as a violation of the Establishment Clause as it promotes an endorsement of religion within public education.