No, it doesn't unless modified by the legislature, where the power resides, not with the courts. There is no Right to same sex marriage, it is something that can be allowed by legislatures.
No one ever said there was.
In fact, theres no such thing as same-sex marriage.
All states have but one marriage law, contract law, marriage contracts that both same- and opposite-sex couples are eligible to enter into.
The 14th Amendment requires the states to allow American citizens, residents of the states, access to all of a states laws, including marriage law. When the states contrive un-Constitutional measures to prevent same-sex couples from accessing the marriage law theyre eligible to participate in, such as Utahs Amendment 3, those states violate the Equal Protection Clause of the 14th Amendment:
[T]he Supreme Court has considered analogous questions that involve the tension between these two values in other cases. See, e.g., Loving v. Virginia, 388 U.S. 1 (1967) (balancing the states right to regulate marriage against the individuals right to equal protection and due process under the law). In these cases, the Court has held that the Fourteenth Amendment requires that individual rights take precedence over states rights where these two interests are in conflict. See id. at 7 (holding that a states power to regulate marriage is limited by the Fourteenth Amendment).
Utahs Amendment 3, as with other similar measures invalidated by the courts, lacks a rational basis, lacks compelling evidence in support, and pursues no proper legislative end; it seeks only to make gay Americans different from everyone else, in violation of the 14th Amendment.