Does the Greek word "arsenokoitai" really refer to homosexuality?
That word is an unusual word. It’s a new word; we don’t know of any other instances of the word until Paul coins the word in 1 Corinthians 6 and 1 Timothy 1. It’s a compound word: “arsen” means man and “koite” or “koitas” or “koitai”—depending on a verb or a noun—means bed. It’s men who bed with other men.
It’s quite clear that Paul has coined this word from Leviticus 18 and 20. Even if you don’t know any Greek, you could find online or pull up the Greek transliteration of these two verses. Look at the Septuagint, which is the Greek translation of the Hebrew Bible, and then just look at the Greek for the necessary passage in Leviticus 18 and Leviticus 20 and you'll see there (and in fact the words are right next to each other in Leviticus 20) this word for man (“arsen”) and the word for bed (“koitai” or “koite”).
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Paul is quite deliberately pulling from the Torah to make this new word. So he has in his mind all that was written in the Old Testament. And that’s part of the reason why we can’t just say, “Well Leviticus has a lot of strange things and so we shouldn’t pay attention to Leviticus.” Because Paul is explicitly drawing this teaching into the New Testament to coin this word which, given the context in Leviticus and how it’s used elsewhere after the New Testament, means men having sex with other men. And there is no real other interpretation that makes the best sense of the evidence both in the early Christian literature and especially in the Old Testament.
How to respond biblically to the question of natural-born homosexuality with grace and truth.
That‘s a fair question, though it’s a question that would have been strange to anyone in the biblical world.
A thoughtful answer to a question that we all—sooner or later—may need to face, and how to honor Christ with our response.
It's just not accurate to say that what we are seeing now as expressions of homosexuality were completely unknown to the biblical authors.