I think the proponents of same sex marriage would respond that what they are trying to do is not prevent the government from regulating behavior (which, as you say, is the essence of lawmaking), but prevent the government from treating persons in same sex relationships differently from persons in opposite sex relationships. The key question is whether the government can discriminate based on sexual orientation.
In the U.S., the Equal Protection Clause of the 14th Amendment prohibits a state from denying any person "the equal protection of the laws." This would appear to forbid discrimination based on any characteristic at all. Of course, governments make distinctions based on people's characteristics all the time. These range from the mundane (persons with poor vision can be required to wear corrective lenses while driving) to the highly consequential (American citizens of Japanese ancestry can be interred during a war against Japan).
The history of the Equal Protection Clause, then, is a long and messy process of sorting out what kinds of discrimination are permissible and what aren't. And there has been a lot of movement, especially during last few decades. Consider that during the lifetime of George Takei, the EPC was interpreted to permit the internment of his family (Korematsu v. United States, 1944) and giving him the right (in California) to marry his husband of the same sex (Hollingsworth v. Perry, 2013).
The current standard is that that you can discriminate on a basis other than certain "suspect" classifications if the discrimination is reasonably related to a legitimate government interest.
In the recent Windsor case, the Supreme Court concluded that the federal Defense of Marriage Act failed to meet even that minimal standard. Many lower courts have invalidated state same sex marriage bans on the same basis.
You can argue that it is the place of the legislature to decide what classifications it can use in lawmaking, but that raises two problems. First, the clause is part of the Constitution and it must impose some limit on the power of state legislatures, and there needs to be a process for interpreting what that limit is. Second, history has shown that the political process doesn't always protect unpopular minorities (though there are also good arguments that courts are not that much better).