Yes, I believe that the institution of marriage first existed in the Garden of Eden between the two humans God fashioned by hand. Marriage is not an invention of society but is rooted in the natural order. Governments don't create marriages but recognize legally the natural institution that has existed from the beginning. The purpose for the first marriage was multi-faceted: companionship, cooperation, pleasure, and procreation. Those purposes are timeless and, in fact, they overlap greatly in the lives of those who become "one flesh" (Gen. 2:24). God blessed the first man and woman and commanded them to "Be fruitful and multiply; fill the earth and subdue it" (Gen. 1:28). God had created them with unique and complementary sexual organs that would allow them to accomplish this very objective. Both a male and a female human are required for producing a new human life after their kind, and bearing children (the next generation) is a fundamental reality of marriage, exceptional cases not withstanding. This is the God-ordained, natural order of our world, and mortals cannot change that reality.
A human society cannot survive without the natural sexual relations of males with females. If only unnatural sexual relations are pursued (e.g., males with males or females with females), the society will perish in a generation. These truths are undeniable. Furthermore, a human society cannot thrive without marriages of men with women (the natural order) because the building block of a stable and enduring society is a family unit founded on a lifetime pairing of a man and a woman (i.e., marriage). Male and female marriage relationships not only produce the next generation but also provide the best environment for rearing children.
From the beginning, every civilization of mankind that has survived and thrived has followed this natural order. Those today who clamor for same-sex marriage seek to redefine what is natural, but passionate words and the desires of many (even the majority) do not change what is abominable into something virtuous. The natural order remains immutable. As Paul wrote in Romans 1:22-29 -
"Professing to be wise, they became fools, and changed the glory of the incorruptible God into an image made like corruptible man-and birds and four-footed animals and creeping things. Therefore God also gave them up to uncleanness, in the lusts of their hearts, to dishonor their bodies among themselves, who exchanged the truth of God for the lie, and worshiped and served the creature rather than the Creator, who is blessed forever, Amen. For this reason God gave them up to vile passions. For even their women exchanged the natural use for what is against nature. Likewise also the men, leaving the natural use of the woman, burned in their lust for one another, men with men committing what is shameful, and receiving in themselves the penalty of their error which was due. And even as they did not like to retain God in their knowledge, God gave them over to a debased mind, to do those things which are not fitting; being filled with all unrighteousness, sexual immorality..."
For a society or government to interfere with the God-ordained reality will only lead to harm. America will learn this truth eventually, though it will be a painful lesson. "Righteousness exalts a nation, but sin is a reproach to any people" (Prov. 14:34).