In short, Nietzsche's argument is not valid at all. It was however, a very common belief, and to some extent still is. I recently answered another question about what role Christianity played in the "fall" of Rome. That being said I am not very familiar with why Nietzsche was so adamantly opposed to Christianity. Though if I'm not mistaken he did view Jesus Christ, the person, as kind of version of his idea of the Übermensch.

Anyway that being said, as I discussed in my linked answer, many people argue that Christianity was responsible for the Roman Empire's "fall" because it weakened the military. But this is entirely untrue, as the Roman military continued to perform well up to the last days of the Empire in the west. There were several factors that led to the "fall" of the Roman Empire, namely the economy. Though it should be noted that the Roman Empire in the west never "fell" in the truest sense of the word, rather it underwent a period of transition that led into the early middle ages.