Mr Trump has accused her team of "attacking" the faith at an event in Florida on Wednesday.
The leaked comments appear to have been written in 2011 and involve a conversation between Mrs Clinton, her communications director Jennifer Palmieri and a think tank.
Allegedly Ms Palmieri makes fun of media mogul Rupert Murdoch for raising his children Catholic.
The Republican House Speaker, Paul Ryan, told the BBC the comments were "staggering."
The email exchange was published this week as part of thousands of emails that appeared on the WikiLeaks whistleblower and leaked documents site.
The message is between Ms Palmieri and John Halpin, a senior fellow at the Center for American Progress, a think tank with White House links.
Mr Halpin, a Catholic himself, said: "(Rupert) Murdoch baptised his kids in Jordan where John the Baptist baptised Jesus."
He went on: "the most powerful elements of the conservative movement are all Catholic.
"They must be attracted to the systematic thought and severely backwards gender relations and must be totally unaware of Christian democracy."
Ms Palmieri appears to have responded that Catholicism "is the most socially acceptable politically conservative religion".
Mr Trump said Mrs Clinton's team had been "viciously attacking Catholics and evangelicals".