Morrison, a Pentecostal Christian, accused Shorten, a Catholic before converting to his second wife's Anglican faith, of a "desperate, cheap shot" ahead of elections on Saturday by challenging the prime minister to say whether he believed gays went to hell.
Morrison said he did not believe gays went to hell, after failing to directly answer the same question from a journalist a day earlier.
"I'm not running for pope, I'm running for prime minister," Morrison told reporters. "So ... theological questions, you can leave at the door."
Australian political leaders' religious views are rarely raised in election campaigns, which have long been regarded as a strictly secular argument over who should govern.
But nine prominent Christian church leaders wrote to both leaders this week demanding protections for religious beliefs and freedom of speech after Australian rugby union team star Israel Folau, the son of a Pentecostal preacher, was found guilty by the sport's administration last week of breaching the sport's code of conduct by using social media to say gays were damned to hell.
While Morrison is a centrist, his opposition to gay marriage was out of step with the 62% of voters who supported gay marriage.
Shorten attacked Morrison for failing to address the theological fate of homosexuals when questioned on Monday.
"I cannot believe that the prime minister has not immediately said that gay people will not go to hell," he told reporters.
Also Tuesday, police charged 15 Greenpeace activists following a protest on Sydney Harbor Bridge demanding climate change action.
Labor has pledged to reduce Australia's greenhouse gas emissions by 45% below 2005 levels by 2030. The government has committed to reduce emissions by 26% to 28% in the same time frame.
Three activists managed to climb the bridge and attach themselves by ropes, dangling from the bridge.
They unfurled banners reading "100% Renewables" and "Make Coal History."
They were later lowered into a police boat in the harbor and arrested, police said.
Greenpeace Australia-Pacific chief executive David Ritter called on Morrison to declare a "climate emergency" before the elections.
But Morrison's only response was to ask the protesters to show consideration for the bridge users, after peak hour traffic was slowed by the demonstration.
"I'm passionate about a lot of things, but I don't want to stop traffic on the Sydney Harbor Bridge because that would be inconsiderate to my fellow Australians," he said.