American Christians who are most preoccupied with sexual morality tend focus on homosexuality, pornography, sex before marriage, etc. Except for the Catholic Church, we don't hear much about divorce and remarriage (and even American Catholic leaders don't say much that is covered in the media). Why? Some Christians do argue against divorce, but even they don't claim that remarriage is a sin. According to the Biblical texts, though, it probably is.
There is more than a bit of hypocrisy in how conservative Christians in America invest so much time, effort, and money into limiting others' freedom (gay marriage, pornography, contraceptives) while ignoring biblical injunctions against behavior that they not only benefit from, but engage in at higher rates than other groups: divorce. We can add remarriage to the mix: conservative Christians don't just accept divorce despite biblical condemnations, but they also accept remarriage after divorce despite the fact that there is more criticism of both in the Bible than there is of their usual targets (homosexuality, pornography, contraception, abortion).
Divorce & Remarriage in the Bible
Here are some relevant passages:
It was also said, "Whoever divorces his wife, let him give her a certificate of divorce." But I say to you that every one who divorces his wife, except on the grounds of unchastity [porneia], makes her commit adultery [moicheia] and whoever marries a divorced woman commits adultery. [Matt 5:31-32]
And Pharisees came up to him and tested him by asking, "Is it lawful to divorce one's wife for any cause?" He answered, "Have you not read that he who made them from 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'? So they are no longer two but one. What therefore God has joined together let no man put asunder." They said to him, "Why then did Moses command one to give a certificate of divorce, and to put her away?" He said to them, "For your hardness of heart Moses allowed you to divorce your wives, but from the beginning it was not so. And I say to you: whoever divorces his wife, except for unchastity [porneia], and marries another, commits adultery; and he who marries a divorced woman commits adultery." [Matthew 19:3-6, Jesus quoted Genesis 2:24]
To the married I give charge, not I but the Lord, that the wife should not separate from her husband (but if she does, let her remain single or else be reconciled to her husband) and that the husband should not divorce his wife. [1 Corinthians 7:10-11]
Luke has the same statement against divorce and remarriage as Matthew 5, but leaves out the exception for unchastity. Mark has the same story as Matthew 19, but again the exception for unchastity is absent. Was this exception created by Matthew and thus didn't exist in the original traditions? This exception would be unhelpful for most Christians because the original Greek word porneia doesn't simply mean "unfaithfulness" (i.e., adultery), but situations where the original marriage was never really valid in the first place.
The word usually used for adultery is moicheia so porneia should probably be translated not as mere "adultery" or even "fornication" but rather "illicit sexual conduct." In this context, "licit" or "lawful" would be controlled by the Jewish laws on sexuality in Leviticus. Therefore, this "exception" arguably refers to situations categorically prohibited in Leviticus., like incestuous relationships. Since such couples are never truly married to begin with, which means there is no actual divorce or re-marriage.
Divorce & Remarriage in America
Let's review: both Paul and Jesus condemn divorce and remarriage. Divorce, if necessary, is acceptable — but remarriage either never is or is only acceptable if there is good reason to believe that the original marriage was never a truly valid union to begin with. Contrast this with the fact that Jesus never actually mentions homosexuality — only Paul has anything to say specifically in the matter.
Given this, one would think that those who claim to worry about the state of sexual morality in America and who also claim the Bible is the absolute, inerrant, and infallible Word of God to be at least as outspoken about remarriage after divorce as they are about homosexuality — especially when we remember that we can find far more remarriage in America and positive images of remarriage in the media than we will of homosexuality.
Now, I'm not saying that such Christians should be silent on subjects like homosexuality — on the contrary, we should expect that they would continue to voice their opposition to anything that would make homosexuality seem acceptable and appropriate. However, the fact that we see these same people saying nothing about remarriage — and in many cases remarry themselves! — indicates that they aren't what they claim to be.
Most likely they are, contrary to their statements, picking and choosing what sorts of biblical laws and values they wish to enforce. It isn't simply a matter of emphasizing "moral laws" which continue to be in effect while dismissing "ceremonial laws" that are no longer relevant — there is an argument for such an approach, but remarriage is as much a "moral law" issue as homosexuality.
Technically the Catholic Church is the main exception — and I say "technically" because while the prohibition on remarriage remains, we don't see Catholic leaders speaking out on it like they do on homosexuality. A lot of time and money is invested in opposing things like laws that might allow gay marriage, but not in changing the laws to prevent remarriage for straight couples after divorce. This is hypocritical, but we shouldn't be surprised at it because, in practice, getting an "annulment" is nearly as easy for Catholics as getting a divorce is. The technical prohibition on divorce is not honored in reality to the degree one would expect from an institution that is truly holding to moral principles.