It isn't about "stability". It is about what happens if there are children and this committed relationship dissolves. How will the children's interests be protected? How will it be decided who has custody? What about inheritance in the case of the death of both parents? Yes, part of it is having society accept them as a married couple, but more importantly are the legal protections that need to be afforded to them.
"It is about what happens if there are children and this committed relationship dissolves."
That's easy. If one of the partners is a biologial parent, that person gets custody and has all parental rights/responsibilities.
Homosexual couples shouldn't be allowed to adopt. If adoption is 'desired,' then one of the members of that 'couple' should adopt the child. A break up would not affect the rights and responsibilities of the single adoptive parent.