There were many critics of Gibbon. He was not the first to put forward a declension narrative, of course. Many before him in the wake of the Reformation had done so (folks like Spinoza, though Gibbon calls him 'impious'). Likewise, when Alaric attacked Rome with the Visigothic army in August of 410, pagans blamed the Christians for the decline of Rome - this is the milieu in which St. Augustine wrote his famous De Civitate Dei (On the City of God).

At any rate, the critics of Gibbon basically said that he was just serving up re-heated skepticism from antiquity. Henry Davis, in his Examination, compares Gibbon to Julian the Apostate and Celsus. Now, whether they were attacking Gibbon because he was embodying the upward mobility of the impious or because they actually thought his arguments were simply aping long-confuted enemies of Christianity is a more difficult question.

There's a good introduction to this in David Womersley's Gibbon and the 'Watchmen of the Holy City': The Historian and His Reputation (Oxford: Oxford University Press, 2002).