No, Christians do not believe in ethical relativism.
I'd first like to address the slavery comment in your question. You said:
God was not able to provide a moral code that slavery was evil, but
using reason man was able to arrive to that conclusion.
It's quite a jump to say that God did not provide such a moral code, all the way to saying that God could not provide such a moral code. Simply because he did not, does not mean he could not. What God is able to do is not simply limited to what he did do.
You might object that he could have, but he didn't, and thus he implicitly supports slavery. Again, this is another jump that simply cannot be made. The Bible is mostly silent on it's approval or disapproval of the institution, and at best, provides more restrictive ways for masters to treat their slaves, which actually added some accountability on the part of the masters, as opposed to allowing the masters to do whatever they want. In other words, the Biblical law surrounding the institution of slavery was given to protect the rights of the slaves, more than it was given to protect the rights of the slaveholders.
It's also worth noting that slavery in the form in which it existed in Biblical times was a far cry from the slavery that existed in the United States in the 18th and 19th centuries, and is still a far cry from the slavery that continues to exist in other parts of the world, but that's another question entirely.
For Christians, the law that is laid out in Scripture is the moral law. These are God's commands to people today, just as they were God's commands to people back in Israel. These laws are eternal, and Jesus himself stated that the law doesn't change (Matthew 5:18)
For truly, I say to you, until heaven and earth pass away, not an
iota, not a dot, will pass from the Law until all is accomplished.
But the heart of the question might be addressed better by answering the question: "What is the purpose of the law?"
Biblically, the purpose of the law is to reveal our sin. Paul says this in Romans 3:19-20:
Now we know that whatever the law says it speaks to those who are
under the law, so that every mouth may be stopped, and the whole world
may be held accountable to God. For by works of the law no human
being will be justified in his sight, since through the law comes
knowledge of sin.
More than this, when we break one law, we are guilty of breaking the entire thing. This means that regardless of your position on controversial topics such as homosexuality or a woman's right to her own body, there's almost certainly a law you can find that you've broken at some point, and in doing so, you've broken the entire law. In other words, God demands perfection (James 2:10):
For whoever keeps the whole law but fails in one point has become
accountable for all of it.
So the purpose of the law isn't that we would use it to excuse ourselves or point out that we've followed it our entire life. It's to point out that we haven't. This isn't an accusation. It's a diagnostic fact. If you want to take it further, you can look at the story of the Rich Young Ruler, who believed that he deserved something from God because of all the stuff he did (Mark 10:17-22):
And as he was setting out on his journey, a man ran up and knelt
before him and asked him, “Good Teacher, what must I do to inherit
eternal life?” And Jesus said to him, “Why do you call me good? No one
is good except God alone. You know the commandments: ‘Do not murder,
Do not commit adultery, Do not steal, Do not bear false witness, Do
not defraud, Honor your father and mother.’” And he said to him,
“Teacher, all these I have kept from my youth.” And Jesus, looking at
him, loved him, and said to him, “You lack one thing: go, sell all
that you have and give to the poor, and you will have treasure in
heaven; and come, follow me.” Disheartened by the saying, he went away
sorrowful, for he had great possessions.
So what did Jesus do? He just made the law harder. (He did this alot, by the way... see how he extended murder and adultery). His point is this: you've failed. Your righteousness is soiled, and you don't meet God's perfect demands.
So when we see that we've broken the law, the knowledge of that is supposed to drive us to Christ, who followed the law in it's entirety, fulfilling the law, and thus being the only one who satisfied God's demands. Since we see in scripture, in Romans 6:23, that "the wages of sin is death", and we see that Jesus died without sin, then Christ's death must have been for something, and we trust God that it's to pay for our trespasses, since he had none of his own to pay for (2 Corinthians 5:21):
For our sake he made him to be sin who knew no sin, so that in him we
might become the righteousness of God.
So to sum up, I suppose my answer to this question is, in some sense, "who cares?" The point of the moral code was never so that we could point at someone else and show them that they are breaking this or that law, but rather that we could point at our own hearts and see that we have broken the law ourselves. It's purpose is to overwhelm us with the understanding that there's no possible way we can obey all of these laws perfectly, and in attempting to do so, we see just how depraved we really are, and then turn in desperation to God, who will give grace to anyone who asks.
So do Christians believe in a single moral law? Yes. Should we bicker about the details of the moral law? Probably not. It's not productive. The point of the law is to show our need for a savior.
For more on this, see my answer to another question here.