The United States Constitution's First Amendment has been determined to mean "free expression". So gay rights advocates argue that being gay is free expression. The 14th Amendment says there's an "equal protection" under the law. Thus gay rights advocates believe they must be allowed the same rights and privileges as any other citizen.
However, when it comes to religion, that is protected under the 1st Amendment too. Thus a religion has the right to choose who is to be a part of its membership. It's like the right to free speech cannot infringe upon the right to peacefully assemble. The LDS Church has every right to refuse membership to those who are gay and it's inappropriate for gay rights advocates to try and force the LDS Church to change.
So why then does the LDS Church not allow gay members? First, is that the definition the Church uses for one who is gay is that of "one who has sexual relations with someone of the same sex". For the Church this is a sin that violates the law of chastity. The LDS Church is not designed for everyone, only for those who are wanting to go to the highest degree of glorification after death. In the LDS theology this means that members are striving to become like their Heavenly Father, who is a glorified man. As his children we are in the image of our Heavenly Father AND our Heavenly Mother. Exaltation, as it is called, is therefore not for people who are gay and is not even for people who are single. It is for those who are one man and one woman sealed in marriage.
Does God make people gay from birth? The answer is no! Our Spirit bodies that Heavenly Father and Heavenly Mother created are perfect, such that the natural inclination, as a Spirit body, is toward that sex to which one is designed; there is no error on the part of God. But then there is this natural agency that is given to mankind, which allows an individual to go contrary to their natures; but this agency does not change the base law of creation. In mortality the Fall of Adam brought death to our physical bodies this means that there are imperfections that can and do occur with the creation of the mortal body. I didn't learn of my sexual preference until I hit puberty, though I was culturally conditioned on top of that to prefer one of the opposite sex. But if there is a gay gene that can be found it still is not enough proof for a religion to alter their theology, teachings, and membership admittance.