Marriage is a covenant, or contract, between a man and a woman concerning their continuing relationship. God used this marital analogy in the Bible to symbolize his relationship with Israel (such as in Ezekiel 16 and Jeremiah 3), and for Christ and the Church (as in Ephesians 5:22-33 and Revelation 19:6-8). Here free will is essential for the contract in question, since Israel (Exodus 19:3-9, 24:3-8) and later the Church chose to be in union with God.
Consensual agreement should be involved before a marriage license is granted. True, the Old Testament describes cases of arranged weddings, such as (in particular) that of Rebekah and Isaac. It does appear, however, that Rebekah did have the ability to say "no," but she did choose to go to Isaac, sight unseen, as an act of faith (see Genesis 24:53 - 59, 64 - 67, especially verse 58). Notice that mutual consent was involved on both sides here at some level.
A primary requirement in making an acceptable marriage before God concerns consent and contract when adults wed each other. Before the sexual union takes place (Genesis 2:24), there has to be a covenant (agreement or license) first. An important consequence of this view is that any religion or culture's ceremony is valid intrinsically in God's sight, whether or not the couples are Christians then or even later.
A non-Christian couple who later convert should not think that because their wedding was performed within a false religion tradition, such as Hinduism, that they are free to divorce as they may wish at some future time.
A private agreement
A marital relationship using a private contract is not sinful based on the Bible. It should be witnessed by two or more parties so it is harder for the participants to deny that the wedding occurred (cf. see the principle of Deuteronomy 17:6). It appears that a state-granted marriage license was relatively uncommon until the late nineteenth century or early twentieth century, at least in the United States.
Although a marrying couple can choose to wed by private contract, which was common before modern governments took over the process from churches or the individual families in question, there are no practical advantages gained by using such an irregular procedure.
Clearly, God delegates to human authorities the power to bind and loose matrimony when they are not violating His law (cf. the principles found in Romans 13:1-7, Matthew 18:18-19, 16:19). Hence, it does not matter whether a couple is wed by a church, by a secular government, by a synagogue, by a Muslim imam, a pagan priest, or a Hindu cleric. They are all considered binding in God's eyes, unless it intrinsically violates His law for some reason, such as homosexual men marrying each other, a man taking an additional wife in a polygamous relationship, etc.
If someone gets married by the government and - or church, they should petition and ask those same authorities to end their relationship publicly. A couple should patiently wait out the legal process before marrying (and having sex) or divorcing. People should not just whimsically decree a 'beginning' or 'end' to their relationship on their own, without having the contract ratified or nullified formally by a publicly verifiable process. The "legalities" do matter in God's sight.
An important passage related to this question concerns Christ's general condemnation of divorce (except for adultery) in Matthew 19:3 - 12, when He was debating the Pharisees’ citation of the Old Testament’s relatively easy divorce law. Since divorce involves the nullification of the marriage contract that God approved when a couple got married, it is wrong to break it except under very limited circumstances.
In the Old Testament, since Israel was a union of church and state, there was a formal procedure for legally ending a relationship between a husband and wife. The husband had to give his wife a paper signifying that he had divorced her. This would be akin to ending a private contract today. Notice the procedure laid out in the book of Deuteronomy, which Jesus dealt with when debating with the Pharisees about divorce.
1. When a man has taken a wife and married her, and it comes to pass that she finds no favor in his eyes because he has found some uncleanness in her, then let him write her a bill of divorce and put it in her hand, and send her out of his house (Deuteronomy 24:1, HBFV, see also verses 2 - 4).
In conclusion, it would be best for couples who wish to be wed to seek to obtain 'the magic piece of paper' or official marriage license. Not only will this fulfill principles in the Bible, it will make it easier for the wife and children to claim and receive any government help should the husband become disabled, killed, or simply abandon the family.
Excerpts from answers by Eric Snow
One of BibleStudy.org's Email Evangelists