Georgia, Ohio, North Dakota, Utah, Virginia, Wisconsin, Idaho, North Carolina, Arizona, Kansas, South Carolina, Florida, Alabama, Texas, Michigan, Arkansas, Kentucky, Louisiana, South Dakota, Nebraska all passed laws that restricted marriage to one man and one woman, and also barred any other type of civil union. (By now, most have been found unconstitutional.) Typical language is "A legal status identical or substantially similar to that of marriage for unmarried individuals shall not be valid or recognized."
List of U.S. state constitutional amendments banning same-sex unions by type - Wikipedia, the free encyclopedia
While the handful of libertarian-leaners don't care about SSM, it's pretty clear that many conservatives do not want same-sex civil unions.
I also think your purported fix is absurd, and fools no one. You want to confer the exact same rights, powers, abilities and meaning to a same-sex civil union, as long as they don't refer to it with the word "marriage?" Great. Does that mean I can snort cocaine, as long as I refer to it as "Bolivian Marching Powder"? Can I solicit a prostitute, as long as I call it "walking the dog?"
And in terms of government staying out: You can already perform a legally meaningless ritual symbolizing commitment with as many people of whatever gender you want. Utah tried to outlaw such rituals recently (to prevent legally meaningless polygamist marriages) and was shot down in the courts. But...
Do you not want your spouse to visit you in the hospital? Should your spouse not be empowered to make medical decisions for you, if you are incapacitated? If you divorce in a contentious manner, should the state play no role whatsoever as a neutral arbitrator? Should the surviving spouses of a soldier who is KIA be cut adrift?
The reality is that we don't live in band-level societies anymore, with informal laws and information institutions. Modern life is too complex, too impersonal, too formalized to merely say "government should stay out of marriage."