Considering both biblical and secular traditions of marriage, what seems to be required for a valid marriage is public acknowledgement and acceptance of the relationship with sex being something that traditionally follows the public recognition.
The New Testament demonstrates strong parallels between marriage and baptism.
As a Baptist, if you're a believer you're expected to make a public declaration (Mat 10:32-33, Luk 12:8-9) of your faith through baptism (Mar 16:16).
In marriage you make a public declaration of your commitment to your spouse.
Jesus' first recorded miracle was at a wedding feast, and in one of his parables he spoke of the foolish virgins waiting for the bridegroom. As someone has already mentioned he spoke of the Samaritan woman as not being married to the man she was currently with. There's plenty of scripture that compares marriage with the church's and individual's relationship with Christ, and nowhere is the suggestion that either is a hidden thing kept out of public knowledge.
I'd say sex without public recognition of a relationship is not marriage, but on the other hand public recognition doesn't need to be as elaborate as many people make it.
While there is mention in the Bible of abiding by local laws (eg render under Caesar what is Caesar's), in many countries there's no law that states that you are required to hold a marriage certificate to be accepted as a couple. It certainly provides legal recognition, but comparing a registry office marriage with no friends or family present to a celebration with them present, but with no official paperwork, I think in some ways the latter might actually represent a more authentic marriage in the eyes of God, unless there were some reason that family would not support the marriage, (however from a Christian perspective even if blood relatives don't support the marriage there's still the church family).
Jesus was extremely clear that divorce for any reason other than sexual immorality (Mat 5:32) on the part of a spouse is adultery, so to suggest sex with good intent is "marriage in God's eyes", it would also have to follow that failure to follow up on that good intent makes both adulterers.
Paul speaks of marriage to avoid inflamed passions leading to sin (1 Cor. 7), but he also urges restraint even in marriage, as to him, Christ should be the supreme centre of our attention. I don't think Paul was suggesting married people should not enjoy sex, but more that sex should not be such a preoccupation that there's no time left for Christ.