Do homosexuals have a right to marry according to the international human rights standards? Not explicitly. They do not even have the explicit right to be homosexual, but jurisprudence has established that homosexuals should not be discriminated. First of all, all human beings, whatever their convictions, practices, behavior etc., have the same rights. So killing or torturing or arbitrarily imprisoning people is always wrong. And if this is done because these people are homosexuals or something else is irrelevant.
Homosexuality is also protected by the right to privacy. Article 12 of the Universal Declaration of Human Rights:
No one shall be subjected to arbitrary interference with his privacy, family, home or correspondence… Everyone has the right to the protection of the law against such interference or attacks.
And nothing is as private as sexuality.
Regarding the right to marry, article 16 states:
Men and women of full age, without any limitation due to race, nationality or religion, have the right to marry and to found a family. They are entitled to equal rights as to marriage, during marriage and at its dissolution. Marriage shall be entered into only with the free and full consent of the intending spouses. The family is the natural and fundamental group unit of society and is entitled to protection by society and the State.
Sexual orientation is not mentioned as an unwarranted limitation. So the use of this article is a weak defense of gay marriage. But Article 2 states:
Everyone is entitled to all the rights and freedoms set forth in this Declaration, without distinction of any kind, such as race, colour, sex, language, religion, political or other opinion, national or social origin, property, birth or other status.
The “such as” clause signifies that there may be other types of unwarranted distinction. See also Article 7:
All are equal before the law and are entitled without any discrimination to equal protection of the law. All are entitled to equal protection against any discrimination in violation of this Declaration and against any incitement to such discrimination.
And, as pictured in the cartoon: gay marriage is not necessarily a threat to the institution of marriage. A gay couple can be as serious about marriage and about raisong children as a heterosexual couple. Those of us who care about the importance of marriage have much graver threats to deal with.