It's not, see Lawrence v. Texas (same-sex relationships) and Obergefell v. Hodges (same-sex marriages).

You should honestly just read Griswold [0]. It is a strikingly conservative opinion if, for example, you think privacy is also about "the government has to show a compelling governmental interest to regulate firearms". Here are some gems:

"The association of people is not mentioned in the Constitution nor in the Bill of Rights. The right to educate a child in a school of the parents' choice -- whether public or private or parochial -- is also not mentioned. Nor is the right to study any particular subject or any foreign language. Yet the First Amendment has been construed to include certain of those rights."

"In NAACP v. Alabama, 357 U. S. 449, 357 U. S. 462 we protected the 'freedom to associate and privacy in one's associations,' noting that freedom of association was a peripheral First Amendment right. Disclosure of membership lists of a constitutionally valid association, we held, was invalid 'as entailing the likelihood of a substantial restraint upon the exercise by petitioner's members of their right to freedom of association.' Ibid. In other words, the First Amendment has a penumbra where privacy is protected from governmental intrusion."

"The foregoing cases suggest that specific guarantees in the Bill of Rights have penumbras, formed by emanations from those guarantees that help give them life and substance. See Poe v. Ullman, 367 U.S. 497, 516—522, 81 S.Ct. 1752, 6 L.Ed.2d 989 (dissenting opinion). Various guarantees create zones of privacy. The right of association contained in the penumbra of the First Amendment is one, as we have seen. The Third Amendment in its prohibition against the quartering of soldiers 'in any house' in time of peace without the consent of the owner is another facet of that privacy. The Fourth Amendment explicitly affirms the 'right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures.' The Fifth Amendment in its Self-Incrimination Clause enables the citizen to create a zone of privacy which government may not force him to surrender to his detriment. The Ninth Amendment provides: 'The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people.'"

"In presenting the proposed Amendment, Madison said:

'It has been objected also against a bill of rights that, by enumerating particular exceptions to the grant of power, it would disparage those rights which were not placed in that enumeration, and it might follow, by implication, that those rights which were not singled out were intended to be assigned into the hands of the General Government, and were consequently insecure. This is one of the most plausible arguments I have ever heard urged against the admission of a bill of rights into this system, but I conceive that it may be guarded against. I have attempted it, as gentlemen may see by turning to the

Page 381 U. S. 490

last clause of the fourth resolution [the Ninth Amendment]."

"Madison himself had previously pointed out the dangers of inaccuracy resulting from the fact that 'no language is so copious as to supply words and phrases for every complex idea.' The Federalist, No. 37 (Cooke ed.1961) at 236."

I could go on. I urge you to read it.

[0]: https://supreme.justia.com/cases/federal/us/381/479/