No, Natural Law was used by the Ancient Greeks and Ancient Romans, before Christianity.
The Natural Law is exactly that… the law of nature. In regards to same sex marriage, while you do have instances of homosexuality in nature, you cannot have two male organs or two female organs coming together to create new life.
In nature, life is either asexual or sexual. NOTE: While there are some sexual species with both male and female parts and even some species that can naturally change their sex; it takes the male and female sexual reproductive systems to come together to create offspring.
This whole matting process is complex and some species mate for life. “Natural Marriage” is what we call the natural, lifetime matting process of humans.
Furthermore, same sex attraction has always existed and sometimes was even promoted in ancient pagan society. But never would they consider same sex couples to be married, because marriage was the legal way to establish heirs.
Gay men in antiquity would marry women simply to have a legal heir. Even if they only loved men. Alexander the Great was most likely gay, if not bi-sexual. But he surely had homosexual leanings, yet he still had wives for his empire and heirs.
The point is: the idea that marriage is mainly about love is really a modern concept. Historically, marriage was about establishing the legitimacy of children and tying them legally to the father. And it was also about inheritance.
Just look at how many societies, all over the world, had arranged marriages. If the institution of marriage was primarily about love and mainly love, the history would be different. It was more important for a woman to marry someone who would protect her and her children than it was to marry for purely love. Sometimes people were lucky and got both, but it would have been crazy for a woman to marry a “weakling” she loved who couldn’t protect her and the children.
We have to stop thinking with modern ideas… life was very different and frankly alien when you don’t have modern technology.
In regards to a Christian view of the Natural Law in regards to Marriage, Christ later raised the marriage to a Sacrament to provide it further religious significance.
The study of the Natural Law is Philosophy. Theology is the study of God’s Divine Law. While the two are similar, and have some overlap, they are not the same. To a degree, Theology is depends on Philosophy, but Philosophy is independent of Theology.
If you read Thomas Aquinas, you can learn more about the differences.
I pray this is helpful.