A clear majority of Americans support civil unions, and an equal majority oppose same-sex marriage. I am absolutely sure that this is because most heterosexuals simply have no idea just how much of a difference there is between the two relationships, in legal terms.
Be sure to also read the Feministing article linking to this video, as it goes into one of the many true stories behind the hospital visitation claim.
Be sure, also, to see my article on why everybody should support same-sex marriage, and my blog entry explaining why the Fourteenth Amendment demands nothing less.