Please find me anything that says marriage is a right.
Loving v. Virginia
"These statutes also deprive the Lovings of liberty without due process of law in violation of the Due Process Clause of the Fourteenth Amendment. The freedom to marry has long been recognized as one of the vital personal rights essential to the orderly pursuit of happiness by free men."
Zablocki v. Redhail
"In evaluating 245.10 (1), (4), (5) under the Equal Protection Clause, "we must first determine what burden of justification the classification created thereby must meet, by looking to the nature of the classification and the individual interests affected." Since our past decisions make clear that the right to marry is of fundamental importance, and since the classification at issue here significantly interferes with the exercise of that right, we believe that "critical examination" of the state interests advanced in support of the classification is required."
In re MARRIAGE CASES
"If civil marriage were an institution whose only role was to serve the
interests of society, it reasonably could be asserted that the state should have full
authority to decide whether to establish or abolish the institution of marriage (and
any similar institution, such as domestic partnership). In recognizing, however,
that the right to marry is a basic, constitutionally protected civil right..."
Varnum v. Brien
"This case, as with most other civil rights actions before it, implicates
these broad constitutional principles of governing."
<<SNIP>>
"The Iowa General Assembly has recognized the need to address
sexual-orientation-based discrimination by including sexual orientation as a
characteristic protected in the Iowa Civil Rights Act,"
Goodridge v. Department of Public Health
"The history of constitutional law "is the story of the extension of constitutional rights and protections to people once ignored or excluded." United States v. Virginia, 518 U.S. 515, 557 (1996) (construing equal protection clause of the Fourteenth Amendment to prohibit categorical exclusion of women from public military institute). This statement is as true in the area of civil marriage as in any other area of civil rights."
As the judge herself said, it's a privilege.
No she didn't, what the Judge said from the OP was:
"Performing marriage ceremonies is not a duty that I have as the Presiding Judge of a civil district court. It is a right and privilege invested in me under the Family Code. I choose not to exercise it, as many other Judges do not exercise it."
That's saying her performing Civil Marriage ceremonies is a right and a privileged for her under the Texas Code, one that she chooses not to exercise. That is not saying equal access to Civil Marriage laws is a privilege (equal protection is actually an enumerated right in the Constitution, but that having her perform a voluntary service is a privilege.
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