Uhh, no.
The heyday of homosexual influence in the Empire actually coincided in time very neatly with its golden age, as indeed it did in Greece.
When the Empire fell, it had been Christian, with sodomy a capital crime, for over 150 years.
We could argue forever about why Rome fell, but it had little if anything to do with homosexuality.
This ridiculous notion came about because when people think of Rome they think of Nero and Caligula, who are among the best-known Roman emperors simply because they had the most entertaining stories told about them by historians.
You are right in your statement that after Emperor Constantine converted the empire to Christianity the were laws against homosexual act.
I did not say that homosexuality caused the fall of the Roman Empire but my poor wording could cause someone to infer that.
Thanks for the clarifications. All your statements are correct.