how does banning same sex marraige violate the right to privacy law?

I didn’t know there was such a thing as privacy law. It does, however, conflict with separation of church and state. If you question this, try to find someone who is against same-sex marriage who doesn’t use “because it’s in the bible” as the basis of their argument.

There is no explicit right to privacy. It is more of an implicit idea developed over many court cases.

In my personal opinion it does not violate any right to privacy to ban same sex marriage.

From Roe vs Wade

“The Constitution does not explicitly mention any right of privacy. In a line of decisions, however, going back perhaps as far as Union Pacific R. Co. v. Botsford, 141 U.S. 250, 251 (1891), the Court has recognized that a right of personal privacy, or a guarantee of certain areas or zones of privacy, does exist under the Constitution. In varying contexts, the Court or individual Justices have, indeed, found at least the roots of that right in the First Amendment, Stanley v. Georgia, 394 U.S. 557, 564 (1969); in the Fourth and Fifth Amendments, Terry v. Ohio, 392 U.S. 1, 8-9 (1968), Katz v. United States, 389 U.S. 347, 350 (1967), Boyd v. United States, 116 U.S. 616 (1886), see Olmstead v. United States, 277 U.S. 438, 478 (1928) (Brandeis, J., dissenting); in the penumbras of the Bill of Rights, Griswold v. Connecticut, 381 U.S. at 484-485; in the Ninth Amendment, id. at 486 (Goldberg, J., concurring); or in the concept of liberty guaranteed by the first section of the Fourteenth Amendment, see Meyer v. Nebraska, 262 U.S. 390, 399 (1923). These decisions make it clear that only personal rights that can be deemed “fundamental” or “implicit in the concept of ordered liberty,” Palko v. Connecticut, 302 U.S. 319, 325 (1937), are included in this guarantee of personal privacy. They also make it clear that the right has some extension to activities relating to marriage, Loving v. Virginia, 388 U.S. 1, 12 (1967); procreation, Skinner v. Oklahoma, 316 U.S. 535, 541-542 (1942); contraception, Eisenstadt v. Baird, 405 U.S. at 453-454; id. at 460, 463-465 [p153] (WHITE, J., concurring in result); family relationships, Prince v. Massachusetts, 321 U.S. 158, 166 (1944); and childrearing and education, Pierce v. Society of Sisters, 268 U.S. 510, 535 (1925), Meyer v. Nebraska, supra.”

So you would need to make the argument that same sex marriage was a fundamental right, although I am not sure which definition of fundamentality you would use. However, once you had enough to back your claim that the right to same sex marriage was fundamental you would want to show that the law that infringes this right is does not bear bear a necessary relation to a compelling end. Which basically means their reason for banning gay marriage must be very important and banning same sex marriage must be the only way for them to achieve that goal.

Edit:

From Loving v Virginia

“Marriage is one of the “basic civil rights of man,” fundamental to our very existence and survival…. To deny this fundamental freedom on so unsupportable a basis as the racial classifications embodied in these statutes, classifications so directly subversive of the principle of equality at the heart of the Fourteenth Amendment, is surely to deprive all the State’s citizens of liberty without due process of law. The Fourteenth Amendment requires that the freedom of choice to marry not be restricted by invidious racial discrimination. Under our Constitution, the freedom to marry, or not marry, a person of another race resides with the individual and cannot be infringed by the State.”

Thus denying people the right to marry someone of whatever race they choose was deemed unconstitutional. That may help you, but same sex marriage is not the same case and so this is not necessarily applicable. However it also sounds like you paper may not need to be so detailed so you should look up this case.

The thing is in general laws that infringe our rights are allowed to exist. What you seek to prove is that the government has no justification to do so, which requires following a set of rigid rules about when the government can and cannot deny our rights.