Sexual immorality was a leading contributor in the downfall of numerous civilizations most notably being the fall of the Roman Empire.

Think of it this way: by what mechanism could this possibly operate? How precisely might homosexual activity generate the specific problems that various "civilizations" have faced, and which often include foreign invasions, structural economic tensions, political instability, environmental degradation and reduced productivity of farmland, internal cultural tensions, and so on. I see no way in which homosexual activity could actually contribute to one of those in a meaningful way.

You could also point out to people that no serious historian has argued this in at least a century, if ever. To claim that sexual immorality led to the "decline of civilizations" in the past is really fundamentally a moral claim about the present (in other words, "our 'civilization,' by which I mean the things that I like about my society, are going to be destroyed by people doing things I don't like!"), and not an evidence-based argument about the past. There is simply NO evidence whatsoever to support such a claim.

And that is just to critique the question at face value, without even questioning the utility of viewing "civilization" as things that "rise" or "fall": a "civilization" is, after all, not a clearly defined thing. Even the term itself carries with it value judgments about what constitutes "civilized" versus "savage" behavior, what units are appropriate for the study of history and indeed what groups of people actually have history. The term itself is only with great caution used in academic parlance now, so vague and problematic is it.

Edit: Sorry, I first wrote my response as though you had said "empires," and I had to amend it when I saw that the question was about "civilizations." That makes the initial claim you're discussing even more problematic!

There is no plausible way, as /u/agentdcf points out, that homosexuality could cause any kind of the internal and external factors which cause the decline of empires or civilizations; it is a fallacy cooked up by pseudo-historians and passed off as fact.

Furthermore, a lot of these "civilizations" (i.e. the Roman Empire, Hellenic world, etc.) had knowledge of, and in some cases sanctioned, homosexuality and still rose to become world-influencing civilizations. Take for example the classic Greek city-states; Spartans paired younger men with older mentors who also were their lovers, yet they held the important line at Thermopylae, helping turn the tide against the Persians in the 2nd Greco-Persian War. The Sacred Band of Thebes [a core of hard-fighting hoplites comprised exclusively of paired male lovers] was so valiant and so brave that Philip II of Macedon cried when he ordered them to be slaughtered to a man; he ordered a statue erected in their honor after the fact.

These attitudes smack of a permutation of Edward Gibbon's 18th century treatise The History of the Decline and Fall of The Roman Empire (1776) wherein Gibbon argues that Christianity - pacifist, mellow, loving, Christianity - caused Rome to fall. His take on Roman history has been widely discredited but, at the time, he was attacking Christianity (and more specifically Catholicism) more than he was tackling the issue of the disappearance of the Western Roman Empire.

Let's turn this example on its head: the inference here is that heterosexual conduct somehow invokes God's protection - whatever that is - and prevents the chosen people from 'declining'. Then why were the Israelites - who purportedly lived by the Old Testament which proscribes homosexuality - conquered time and again? The Babylonians, the Egyptians, The Assyrians, the Macedonians, and the Romans themselves stomped all over "God's chosen people". The Jewish diaspora was caused by Romans; could the argument be made that the Jews must have been more homosexual than the Romans? It is ludicrous on its face and any serious assertion should be met derision.

What you have with this argument is a misapprehension of history twisted to fit a personal bigotry in order to justify hateful and selfish attitudes. There really is not a meaningful and articulate way to make someone who thinks these things change their mind, mostly because they'll twist facts, scripture, and a whole host of other sources to fit their mindset.