The West Australian MP and former special forces soldier accused Senator Hanson of failing to take responsibility for her actions and hiding behind "childish" material.
"She's revealed in the last week that she doesn't have what it takes to be prime minister of this country," Mr Hastie told News24 on Sunday.
"When I put the pressure on her over the cartoon ... she showed herself to be rattled.
"If she were to be prime minister, is she going to run cartoons about all her political opponents? How is she going to devote her time?
"Is she serious? Does she have a worldview that we can actually understand? These are all legitimate questions for One Nation."
Senator Hanson has brushed off complaints about the "traitor" comments in the cartoon, saying it shouldn't be taken too seriously despite attracting criticism from across the political spectrum.
Mr Hastie has been attacked by One Nation for testifying during the defamation trial of accused war criminal Ben Roberts-Smith, both of whom served in the elite Special Air Service Regiment in Afghanistan.
The Victoria Cross recipient has been charged over five alleged murders involving unarmed detainees during his deployment in Afghanistan between 2009 and 2012.
Roberts-Smith denies the allegations.
Displaying a photo of an Australian prisoner of war being beheaded by Japanese forces during the Second World War, Mr Hastie said Senator Hanson must clarify whether she believed it was acceptable to execute prisoners.
"She's got to come out and make her position very clear," he said.
"What's her standard for battlefield conduct?"
Asked about recently resurfaced comments in which Senator Hanson criticised Prime Minister Anthony Albanese's late mother for living in public housing for decades, Mr Hastie replied: "Let's leave mums out of it."
"This is another indicator with the Australian people that she's not fit to be prime minister," he said.
Trade Minister Don Farrell labelled the comments "appalling".
"There's a certain bitterness in One Nation's approach to the world that I find quite un-Australian," the Labor minister said.
"I think Ms Hanson should apologise.
"She wants dignity and respect from her fellow parliamentarians. Attacking the prime minister's mother like this is not a way to get respect and dignity from your colleagues."
22°C