It's not that you lack the right to conduct a government-approved ceremony and obtain the legal status of a married couple; you can do that. You can even file your federal tax returns as a married couple, as long as you don't have matching genitals. But nowhere in the United States do you have the right to credibly contract for a lifetime marital partnership.
Every state currently allows some form of "no fault" divorce - divorce not based on any wrongdoing of a party, but simply because the parties claim they don't want to be married anymore. Even though the couple may "vow" to remain together until one of them dies, everyone knows these vows have no legal or real-world effect. The marital "contract" is not a contract at all.
Imagine a regular legal contract in which either party could end the agreement by saying he didn't like it anymore. Could the purposes of contract law be served by such a contract? From a law and economics point of view, such an "if-I-feel-like-it" contract would not support the reliability of contracts, and would require an inefficient level of hedging. The legal term for such a contract is an illusory contract - one that doesn't have any legal effect, of which the legal system will take no notice.
Marriage once did have a legal effect - once married, parties could not divorce without a really good reason (physical cruelty, desertion, or adultery). Not coincidentally, marriages were much more likely to be reliable lifetime partnerships. In addition to the legal strictures surrounding marriage, social groups essentially forced couples to stay together or risk social death.
With the nationwide adoption of no-fault divorce and the elimination of the social stigma of divorce, the nature of marriage changed from a genuine contract to an illusory contract. Marriage stopped being the reliable, socially enforced lifetime partnership it had been for generations.
Poly people might be tempted to think of the destruction of socially enforced monogamy as a good thing. Indeed, many people who did not really want to marry were no doubt forced to do so, and forced to stay married, because attractive options did not exist.
However, even poly people must on reflection realize that an important right has been lost: the right to reliably, credibly commit oneself for life. Even those who think polyamory is the best choice for them rarely want to force their lifestyle on others; indeed, they are often some of the most vocal supporters of expanding the right to marry to gay people. Sadly, however, in allowing anyone the right to divorce at will, we have deprived everyone of the right to truly marry.
Sensing that marriage is now an empty institution, some couples have specifically contracted for the rights marriage traditionally gave them (but no longer does). In the California case Diosdado v. Diosdado, 97 Cal.App.4th 470, a husband and wife contracted that if the husband had an affair with another woman, he would pay the wife $50,000 on top of the divorce settlement, and vice versa. The husband did in fact have an affair, but the California court refused to honor the couple's agreement. The strong California public policy of no-fault divorce, the court said, prohibited courts from even enforcing the voluntary contracts of a mature adult couple:
The family law court may not look to fault in dissolving the marriage, dividing property, or ordering support. Yet this agreement attempts to penalize the party who is at fault for having breached the obligation of sexual fidelity, and whose breach provided the basis for terminating the marriage. This penalty is in direct contravention of the public policy underlying no-fault divorce.
That's right: in California, as in other states with a strong no-fault public policy, you can't even voluntarily make a credible promise of marriage and expect it to be honored by the courts.
A few states - Lousiana, Arizona, and Arkansas - allow what is called "covenant marriage," marriage that may only be dissolved on fault grounds. However, couples may not even use covenant marriage to credibly promise lifetime partnership, because either partner may simply relocate to a non-covenant-marriage state and initiate no-fault divorce proceedings there.
To recapitulate, no one in our country truly has the right to marry, in the sense of committing oneself to a partnership for life. It is legitimate to wonder if this deprivation of rights has caused the proliferation of both tattoos and ludicrously expensive wedding ceremonies, as a last-ditch effort for permanence.
Not even mature adults of sound mind, after long reflection, may decide to marry for life. However, there is another right that has been found to be so fundamental that it cannot be taken away, even from those who have demonstrated that they will abuse the right: the right to breed. The right to have children, like the right to marry, is not mentioned in the Constitution, but is interpreted as protected by the implied right to privacy. However, unlike the right to marry, the right to breed has been genuinely preserved. Mothers who starve their children to death, Trammel v. State, 751 N.E.2d 283 (Ind. Ct. App. 2001), and fathers who make no effort to support their children, State v. Talty, 103 Ohio St.3d 177 (2004), may not be restricted in their "fundamental right" to have as many children as they can.
It is sad for all of us that the law protects the right of each person, no matter how cruel or stupid, to create new, needy, suffering human beings; but no person, no matter how mature, is trusted with the right to credibly commit him- or herself to marriage for life.