The short answer is no.

The long answer is nooooooooooooooooooooooooo.

The idea that Christianity was a contributing factor to the fall of Rome mostly stems from Edward Gibbon's The Decline and Fall of the Roman Empire. Though a groundbreaking book for the study of history, and a very sophisticated analysis of the problem for the 18th century, it's very much a product of its time. Specifically, it is looking back from the Enlightenment at that dark and horrible period of the "middle ages" which separated Gibbon from great and glorious Rome. Or, that is roughly how things were understood.

Modern historians talk a lot about what caused the fall of Rome, from political and economic reasons to environmental causes, but the adoption of Christianity is no longer on the table.