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No - Pope Paul described the potential for catastrophe in its first cause: breakdown of morality. Islamic takeover is an effect.

A few lines from the Durants on the fall of Rome:

A great civilization is not conquered from without until it has destroyed itself within.

“What had caused this fall in population? Above all, family limitation. Practiced first by the educated classes, it had not seeped down to a proletariat named for its fertility; by a.d. 100 it had reached the agricultural classes. … Though branded as a crime, infanticide flourished as poverty grew. Sexual excesses may have reduced human fertility; the avoidance or deferment of marriage had a like effect, and the making of eunuchs increased as Oriental customs flowed in to the West.

“The rapidly breeding Germans could not understand the classic culture, did not accept it, did not transmit it; the rapidly breeding Orientals were mostly of a mind to destroy that culture; the Romans, possessing it, sacrificed it to the comforts of sterility. Rome was conquered not by barbarian invasion from without, but by barbarian multiplication within.

“Moral decay contributed to the dissolution. … Men had now, in the middle and upper classes, the means to yield to temptation, and only expediency to restrain them. Urban congestion multiplied contacts and frustrated surveillance; immigration brought together a hundred cultures whose differences rubbed themselves out into indifference. Moral and esthetic standards were lowered by the magnetism of the mass; and sex ran riot in freedom while political liberty decayed.”