Justice Scalia—yes, him—wrote the decision of the court, a majority consisting of everyone other than Justices Thomas and Alito. It relies on the Elections Clause of the Constitution (Art. I, §4, cl. 1), which provides that while states have preliminary control over federal elections, Congress can supersede the states' choices:
The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof; but the Congress may at any time by Law make or alter such Regulations, except as to the places of chusing Senators.And, basically, the Court holds that when the NVRA says the states must "accept and use" the federal form, it must accept and use them as sufficient to register voters:
When Congress legislates with respect to the “Times, Places and Manner” of holding congressional elections, it necessarily displaces some element of a pre-existing legal regime erected by the States. Because the power the Elections Clause confers is none other than the power to preempt, the reasonable assumption is that the statutory text accurately communicates the scope of Congress’s preemptive intent. Moreover, the federalism concerns underlying the presumption in the Supremacy Clause context are somewhat weaker here. Unlike the States’ “historic police powers,” the States’ role in regulating congressional elections—while weighty and worthy of respect—has always existed subject to the express qualification that it “terminates according to federal law.” In sum, there is no compelling reason not to read Elections Clause legislation simply to mean what it says.The majority notes that the state may use information in its possession to refuse to register a voter, but it cannot require the voter to show her papers first. At most, Arizona can ask the federal Elections Assistance Commission (EAC) for permission to amend its form, as Louisiana did in 2012 to require applicants who lack a Louisiana driver’s license, ID card, or Social Security number to attach additional documentation (such as a utility bill) to the completed Federal Form.
We conclude that the fairest reading of the statute is that a state-imposed requirement of evidence of citizenship not required by the Federal Form is “inconsistent with” the NVRA’s mandate that States “accept and use” the Federal Form. If this reading prevails, the Elections Clause requires that Arizona’s rule give way.
Arizona loses. Voters win. What else? How about a solo Thomas dissent!
Justice Thomas relies on Art. I, §2, cl. 1 of the Constitution ("The House of Representatives shall be composed of members chosen every second year by the people of the several states, and the electors in each state shall have the qualifications requisite for electors of the most numerous branch of the state legislature") as his basis for insisting that Arizona could indeed take steps to ensure that voters were citizens, and that that clause was effectively erased by the Court's decision. He and Justice Alito mock the majority's suggestion of pursuing relief via the EAC, given that "the EAC has no members and no current prospects of new members. Offering a nonexistent pathway to administrative relief is an exercise in futility, not constitutional avoidance," but otherwise it's a fairly dry (and history-heavy) read.
Justice Alito, while dissenting, also withholds the rhetorical fireworks this time:
The Court reads an ambiguous federal statute in a way that brushes aside the constitutional authority of the States and produces truly strange results.There are likely three more decision days this term: this Thursday, next Monday, and an additional day next week.
Under the Constitution, the States, not Congress, have the authority to establish the qualifications of voters in elections for Members of Congress. See Art. I, §2, cl. 1 (House); Amdt. 17 (Senate). The States also have the default authority to regulate federal voter registration. See Art. I, §4, cl. 1. Exercising its right to set federal voter qualifications, Arizona, like every other State, permits only U. S. citizens to vote in federal elections, and Arizona has concluded that this requirement cannot be effectively enforced unless applicants for registration are required to provide proof of citizenship. According to the Court, however, the National Voter Registration Act of 1993 (NVRA) deprives Arizona of this authority. I do not think that this is what Congress intended.
11:00 AM PT: As Josh Blackman has flagged, Justice Thomas' dissent contains a footnote which constitutes the first time a Supreme Court justice has cited Bush v Gore as precedent in the 12 1/2 years after the case was decided.