I think the majority of the philosophical arguments around gay marriage and gay adoption have been within the realm of political philosophy. Here, there are two major ideas at work in contemporary society:
- Rights and freedoms-based views
- Natural law based views
From what I gather, there's not too much argument in the rights/freedoms side. If you give people the right to have sex with whom they will and to contract with whom they will, that the state has little interest or position in managing who adopts whom. Or to put it another way, you're going to have to go the full distance and say gay sex is immoral to make a rights argument against legal recognition of gay relationships. Or for gay adoption, you're going to have to show specific harm as to why it among other forms is unacceptable.
Thus, the majority of the argument has been about whether a natural law (here merely meaning an account of society and the family informed by something in nature) would allow for gay marriage and gay adoption.
The primary text I'm aware of making the against allowing gay marriage and gay adoption is Robert George at Princeton Univeristy, Ryan Anderson, and Sherif Gergis, What is Marriage?. I don't know the exact details of the argument but it builds a case that there's a specific social benefit to privileging and enshrining the natural family of man, woman, and children in law and giving it benefits, and that other relationships do not merit this privileging. (Actually, Hegel makes a similar argument in Philosophy of Right "The Family" and such arguments (well or poorly made are pretty common in the history of Western philosophy)).
The argument in terms of natural law in favoring of allowing gay marriage and gay adoption hinges on arguing that the purposes of sex and family have less to do with procreative ability and more to do with creating strong bonds through sex. A primary advocate of this position is John Corvino.