Kavanaugh did not rule the EO was unconstitutional. Kavanaugh ruled it violated a previous Statute, and that Statute could be modified. But, the EO couldn’t take effect due to the Statute—not because of the Constitution or any Amendment.
Justice Kavanaugh disagreed with the majority’s constitutional holding, but he concluded that Trump’s executive order violated a federal statute, 8 U.S.C. §1401(a). “Congress could—consistent with the Fourteenth Amendment—amend §1401(a) or otherwise enact new legislation establishing exceptions to birthright citizenship for children born to foreign citizens unlawfully or temporarily in the country. But Congress has not yet done so.”
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So, effectively, the ruling was 6-3 to deny the implementation of the EO. But, not because of the Constitution in Kavanaugh’s decision.