If having a roof over your head is a fundamental right as well, does that mean the gardener should move into my house?
That's not a very good comparison, I fear. Heterosexuals are free to marry in the same way that homosexuals want to, but that way still involves finding a willing and available partner.
How about we build him his own?
That's the idea. Or, rather, they should be
allowed to build their own and to have them officially recognized, just like as anyone else's, for practical purposes such as applying for job interviews, obtaining documents and the like.
Just like the gay people who have been previously disadvantaged and have not been able to marry...should we give them the heterosexaul marriage institution,
They are allowed to marry heterosexually, but that is not what they want or need.
or give them their own institution?
This is not just about semantics. Many civil rights are in some way or another derivated from official recognition of marriages, leading to an avoidable and unfair overburden to same sex couples who can't legally marry. Perhaps even more important are the social stigmas attached.
Why are gay people not happy with civil unions? Rather fight that to make it what they want it to be, dont take what belongs to another.
It just so happens that demanding recognition of same sex marriage is pretty much the only way of making it (their unions) what they want them to be, due to many rules of various degrees of legal weight.
As for taking what belongs to another... how does that apply? Same sex marriages don't really "take away" anyone from other married couples, do they?