The following is adapted from an email I sent my step-uncle after he argued that legalizing same-sex marriage would lead to legalizing incest, bestiality and polygamy.
In the United States of America, no federal or state law can be based only on moral beliefs.
That does not mean morality does not have a place in public policy — the morality and values of individual elected officials do and should have an impact, to a great extent. When you vote for someone, you also in a way are voting for their values and their worldview.
But our Constitution, and in particular the First, Fifth, and Fourteenth amendments, means that no law can discriminate against a particular group of people only because of moral disapproval of that group. There must be provable harms, backed up by evidence, that such a group or such a group taking part in such an action do to themselves or to others. To state the most obvious of many examples, murder is not only morally reprehensible, but it also harms others.
The Defense of Marriage Act fell, and in due time same-sex marriage bans will fall throughout this country, because there is no compelling evidence of any harms to the LGBT community, to children, or to others in having same-sex people be able to get married, so laws that ban such same-sex marriage are unconstitutional under the Equal Protection Due Process clauses.
The union of two people who have committed to be for one another and stay committed to one another in marriage till death do them part is not somehow outside the realm of human experience. It is who those people are. It is what their sexual orientation is. It is how they were born, and to discriminate on the basis of that simply for reasons of moral disapproval, is unconstitutional.
Now, the slippery slope argument you raise is not actually an argument in and of itself, but an excuse to not actually have an argument. Very few people try to take part in a constitutional argument on the basis of why it would be ok to deny same-sex marriage because there truly is not a compelling one.
Issues of age, desire to marry a family member (incest), number of preferred partners (polygamy), and attraction to animals (bestiality) have never been found, and will never be found, to be subject to heightened scrutiny. Therefore, as long as a local, state or federal government classifies a group based on their age, desire to marry a family member, number of preferred partners or desire to have sex with animals, such laws will only be subject to the rational-basis test. Thus, any law that treats those who wish to marry young or wish to commit incest, polygamy, or bestiality separately must be "rationally related to furthering a legitimate government interest."
As such, all laws against incest, polygamy, marrying as a minor, and bestiality pass this constitutional test. To name very few of the rational reasons in a simplified way, laws against incest and minor marriage stand for reasons of abuse and psychological consequences, laws against polygamy stand for reasons of abuse and psychology, and laws against bestiality stand for reasons of psychological consequences.
Laws against same-sex marriage, meanwhile, violate the rational basis test as well as the tests of intermediate and heightened scrutiny. Which is why DOMA was overruled and the 37 states that currently do not recognize same-sex marriage will soon, thankfully, do so as a result of a SCOTUS ruling in the near future.