I have not seen a single scholarly work which places any form of moral decay, including homosexuality, as a real cause for the fall of Rome.
The most interesting I read was that Rome was simply not a long term sustainable model. Without places to expand and new peoples to plunder, there was no economy. As taxes increased and laws made life more restrictive, people simply left to the outlying territories, which increasingly ignored the dictates of Rome. It also made for insurgency which welcomed the "barbarians" versus defending agaisnt them.
The general studies state the weakening of the emperors, and fossilization into a series of static weak military dictatorships was the political collapse. The economy collapsed due to the above, and an overdependence on slaves. The population declined due to hardships from the prior two, and people emigrating (as said above). These items left Rome unable to defend itself against enemies growing stronger and learning from the Romans, who did not progress.
Lead in all the food and water supply had more to do with the fall of Rome than morality in general and homosexuality in particular.
