I agree that the practice of homosexuality is a sin. They shouldn’t do that. They shouldn’t get married. These are my beliefs. I also believe that everyone should honor their mothers and fathers, and that we should all go the Church on Sundays. In fact, I believe all that the Holy Catholic Church teaches and believes.
But should we have laws that force nonbelievers to follow our beliefs?
Should there be a law against renting rooms at hotels or apartments to unmarried heterosexual couples?
Should all porn be against the law?
Should we imprison men and women who show disrespect for their parents?
What about people who don’t go the Church on Sunday? Should they face a punishment of law?
Where would you draw the line?
This issue is not specifically a Catholic issue. Rather, it is an issue about the natural institution of Marriage (as opposed to the Sacrament). This applies to all peoples because it is an issue that is related to the Natural Moral Law. The reason why it is so important is because the State (according to the Natural Moral Law) exists for the very purpose of defending the institution of Marriage so that human flourishing can happen. In other words, if there was not Marriage there would be no need for the State.
While the other issues you point to are important they do not have the same foundational priority to human civilization. While all sin/vice does damage to human civilization there are some such as lying/murder/adultery (along with all other aspects that damage the human family such as divorce, spousal abuse, contraception, abortion, etc) that when committed stab at the very heart of human civilization.
Part of the reason why this can be difficult to see in our time is that we are so highly influenced by modern and post-modern thought. In these philosophical strains of thought there is a hyper-emphasis on the autonomy of the individual over and against the community. While the individual is important and we should not establish of false dichotomy between the individual and the community. It would be wrong to assert that the individual can flourish without the community. In this way, the institution of Marriage relates back to the individual and then radiates out, again, to the greater community.
A short quippy way of putting the issue is that the human family is the fundamental building-block of the human State. And, the State exists for the very purpose of supporting and sustaining the flourishing of the human family.
So, to answer your initial question, it is not about forcing people who are not Catholic to support and legally enshrine Catholic beliefs. Rather, the Church is also responsible for the defense of the Natural Moral Law, which is common all all men. Thus, on this issue (and most other moral issues that the Church addresses) the issue is not about strictly about a faith response to a moral issues. Rather, the Church is defending the common moral fabric of all humanity that can be discerned without the aid of Divine Revelation. Just because it is defending by the Church does not mean that it is a religious issue. The Church, in this case, is not arguing for the defense of the Sacrament of Matrimony, but is rather arguing for the defense of natural Marriage.