I agree with what the others have written here about a marriage needing to be legitimized through the legal system (because we are commanded to obey the laws of the country), and also because a marriage needs to be a public proclamation. Another verse in the Bible to consider is in
John 4 when Jesus spoke to the woman at the well...
"15The woman said to him, "Sir, give me this water so that I won't get thirsty and have to keep coming here to draw water." 16He told her, "Go, call your husband and come back."
17"I have no husband," she replied.
Jesus said to her, "You are right when you say you have no husband. 18The fact is, you have had five husbands, and the man you now have is not your husband. What you have just said is quite true."
In this verse we clearly see a woman who has been married five times, and is currently living with/ sleeping with a man who is not her husband. Jesus makes the distinction between the men who were her husbands and the man who is not. Even though the woman is living with a man, and joined herself to him in the flesh, he is not her husband.
Some other things along the lines to consider is that the commitment to marriage actually begins with the engagement. Consider Mary and Joseph as an example..... In
Mathew 1 it says "18This is how the birth of Jesus Christ came about
: His mother Mary was pledged to be married to Joseph, but before they came together, she was found to be with child through the Holy Spirit. 19 Because Joseph her husband was a righteous man and did not want to expose her to public disgrace, he had in mind to divorce her quietly." In these verses we find where Joseph was pledged to be married, but at the same time was considered "married" and that he would need to "divorce" Mary. So, by their engagement, they had entered into a marriage covenant. This indicates that marriage begins prior to an actual ceremony, prior to actually living together, and prior to even having sexual relations. In that same chapter of Matthew, we find a couple of verses later where the angel visits Joseph and tells him to not be afraid to take Mary home as his wife. And at the end of the chapter, it says "24When Joseph woke up, he did what the angel of the Lord had commanded him and took Mary home as his wife. 25But he had no union with her until she gave birth to a son. And he gave him the name Jesus." So, we can see from verse 19 that Joseph would have needed to divorce Mary (because he was already her "husband") even though they were technically what we call being engaged or promised to be married.
And finally, if we look at
Matthew 9, one passage on divorce, you can find where it says " 9 I tell you that anyone who divorces his wife, except for marital unfaithfulness, and marries another woman commits adultery." It is saying that a man who divorces for any reason not permitted by God, is not allowed to remarry because then he will be committing adultery.... and why? Because the marriage covenant is still in effect under God. Backing up a few verses.... "Therefore what God has joined together, let man not separate." A legal divorce is not the same as a spiritual divorce. Unless God has permitted the divorce under certain conditions, then the spiritual marriage is still valid and any relationship which follows is adultery.
Matthew 9 also says that "and whoso marrieth her which is put away doth commit adultery." If you marry a woman (or man) who divorced for the wrong reasons, then you also are committing adultery because that person is still spiritually joined to their original spouse. Only God can spiritually separate a marriage, and he has given instructions on when it is permissible. So, I am bringing up this final point about marriage in regards to divorce, in order to show that there is also a distinction that can be made between a legal marriage and a spiritual marriage. This does not negate the need for a legal marriage.
Anyhow, this is my take on what the Bible says about marriage. Marriage begins with the promise/ public pledge to be married. It begins with the uniting of two hearts along with a public proclamation of this pledge. At that point the two are joined. (Unless the one is already joined to another and then it is simply adultery.) After that point, the union is made legal. And from there it can only be separated (divorce) under certain conditions, and only the "innocent" party is held blameless and can remarry. Anything else would be sin.