Question:Does God disapprove of my marriage?
I have been with my (wife) for over 3 years now. I know that a marriage must have a witness for it to be true. The day we wed we were under the canopy of heaven (outside.) We asked God to be a witness to our union. Does God disapprove of my marriage because it is not through the court system?
Marriages in the United States are not done through the court system. The government provides a record of marriages which take place. Courts only get involved when people desire to break their marriage vows. Our government requires documentation that a marriage has taken place for a variety of reasons. There are only a few states left which recognize a common-law marriage, which is what you are attempting to create. The greater majority have discarded their common-law rules because of the mess created by them.
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.
Paul insisted once that funds sent to needy saints be carried by people selected by the local churches. It wasn't that Paul wasn't trustworthy. He was concerned as to how it appeared to people. By having others involved he was "providing honorable things, not only in the sight of the Lord, but also in the sight of men" (II Corinthians 8:21). But in your marriage you are only claiming to be honorable to the Lord. It is a noble thing, but it is only partial. "Respect what is honorable in the sight of all men" (Romans 12:17).
If you are like most, you are avoiding that "piece of paper" because it would serve as witness to a commitment. Though you claim God as your witness, the fact that you don't want it recorded so that man can also see that a marriage has taken place tells me you have no long term plans to remain committed to the woman with whom you currently live. 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.
It is not whether God disapproves of your marriage, it is whether you are honest enough to admit that you are married in a verifiable way. Even in your own note, you appear to hesitate to call the woman you are living with your wife. This doesn't sound like commitment to me.