4 The First Amendment provides in relevant part that "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof." U. S. Const., Amdt. 1. The Religion Clauses apply to the States by incorporation into theFourteenth Amendment.

That's footnote 4 at the bottom of page 4 of the SCotUS ruling, which annotates this text that begins on the same page:

The complaint seeks a declaration that the 1954 Act's addition of the words "under God" violated the Establishment and Free Exercise Clauses of the United States Constitution,4 as well as an injunction against the School District's policy requiring daily recitation of the Pledge.

(p. 3 of the ruling) In other words, Congress did make a law...

Congress revisited the Pledge of Allegiance 12 years later when it amended the text to add the words "underGod." Act of June 14, 1954, ch. 297, 68 Stat. 249. The House Report that accompanied the legislation observed that, "[f]rom the time of our earliest history our peoples and our institutions have reflected the traditional concept that our Nation was founded on a fundamental belief in God." H. R. Rep. No. 1693, 83d Cong., 2d Sess., p. 2 (1954). The resulting text is the Pledge as we know it today: "I pledge allegiance to the Flag of the United States of America, and to the Republic for which it stands, one Nation under God, indivisible, with liberty and justice for all." 4 U. S. C. §4.

Here, by the way, is the 14th Amendment; I believe Section 1, which establishes the "equal protection" concept in law, is the relevant portion.

The Supreme Court ducked the issue of whether the 1954 change to the Pledge violates the Establishment Clause of the First Amendment, by ruling solely on the issue of Newdow's standing to sue on his daughter's behalf... but see Chief Justice Rehnquist's dissent (joined in by O'Connor and Thomas), which begins on page 16.

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As a sidebar, you may be interested in a piece of the original 1942 Pledge legislation, which is quoted at the bottom of page 2 of the ruling. I'll re-quote it here:

That the pledge of allegiance to the flag, "I pledge allegiance to the flag of the United States of America and to the Republic for which it stands, one Nation indivisible, with liberty and justice for all", be rendered by standing with the right hand over the heart; extending the right hand, palm upward, toward the flag at the words "to the flag" and holding this position until the end, when the hand drops to the side. However, civilians will always show full respect to the flag when the pledge is given by merely standing at attention, men removing the headdress. Persons in uniform shall render the military salute.

[emphasis added]