I could argue that God doesn't recognize a piece of paper saying if you're married. I believe in His eyes my girlfriend and I are married, and if we left each other we would commit the sin of divorce. If living in sin with my girlfriend is my worst sin, I'll gladly take it. (It's probably not ...)
Your assertion is flawed on a number of points. First, marriages are created by God, not by you or by the government. "And He answered and said to them, "Have you not read that He who made them at the beginning 'made them male and female,' and said, 'For this reason a man shall leave his father and mother and be joined to his wife, and the two shall become one flesh'? So then, they are no longer two but one flesh. Therefore what God has joined together, let not man separate."" (Matthew 19:4-6). It was not a piece of paper that joined Adam to Eve as husband and wife; it was God. Just because you would like to view yourself as married, it does not make it so.
As with all things, a marriage is created when God's commands are followed. In this case, a marriage exists when a covenant of marriage is made between a man and a woman. "Yet you say, "For what reason?" Because the LORD has been witness between you and the wife of your youth, with whom you have dealt treacherously; yet she is your companion and your wife by covenant" (Malachi 2:14). Without that witnessed covenant, there is no marriage.
As it stands, you claim that you are in a state equivalent to marriage, but you simultaneously claim you are not married and acknowledge that you are committing fornication with a woman. You can't have it both ways. And what happens in a few years when you decide that you no longer "love" this woman and you walk out? Will you not then tell everyone that you weren't really married anyway? If you think that it won't happen, you are deceiving yourself again. Couples who live together have the highest rate of separation. [Source: The State of Our Unions, Rutgers' University]. The average length of time for a cohabitating couple to remain together is only 18 months. [Source: Crouse Presents Dangers of Cohabitation, Taylor University].
Second, God tells us to be obedient to the governing authorities. "Let every soul be subject to the governing authorities. For there is no authority except from God, and the authorities that exist are appointed by God. Therefore whoever resists the authority resists the ordinance of God, and those who resist will bring judgment on themselves" (Romans 13:1-2). If our government requires a marriage license for those wishing to claim they are married, there is no harm caused by following that law. It does no harm to the marriage itself. I suspect that you look down on that "piece of paper" because it would serve as witness to a commitment and you have no long term plans to remain committed to the woman you currently live with. You want to retain an imaginary freedom to walk out whenever you decide you don't like how things are going. You don't want any evidence left behind that you had other intentions earlier.
Third, there is no rating of sins with God. All sins are equally harmful. All sins will keep you out of heaven. Take a look at the mixture of sins found in the following passage: "But the cowardly, unbelieving, abominable, murderers, sexually immoral, sorcerers, idolaters, and all liars shall have their part in the lake which burns with fire and brimstone, which is the second death" (Revelation 21:8). The sin you think is not so bad (sexual immorality or fornication) is listed as equivalent to being a coward (which most people would consider mild) and murder (which most people would consider very bad). The point is that sin is sin. To lie to yourself and say that God will overlook my sin because I don't think is so bad is foolish.