Ms Babbit, an Air Force veteran, was killed by law enforcement after she stormed the US Capitol to try and prevent the certification of Joe Biden’s victory over Donald Trump.
Mr Gosar has demanded to know why the FBI has not released the name of the police officer, who was cleared of wrongdoing in April.
Tim and David Gosar attacked the Arizona congressman’s comments and apologized “on behalf of the actual sane members of our family” on CNN on Thursday.
“Once you lose your focus on the truth, once you become someone that peddles in lies, once you become basically a snake oil salesman, the truth is a really slippery thing to get your arms around,” said Tim Gosar.
“It’s really hard to go back to the truth once you become a pathological liar like Paul has become.”
Representative Gosar was one of 21 House Republicans who voted against awarding all law enforcement officers the highest congressional award for their work in protecting the Capitol during the January insurrection.
Mr Gosar’s sister said on Wednesday that her brother was “partly responsible” for the attack and accused the GOP of trying to “erase what is a very documented insurrection”.
“And to know my brother is a part of that is not surprising to me, but it’s something that - it intensifies the frustration about the erasure they’re trying to enact,” she said.
It is not the first time that Mr Gosar’s family have publicly criticized him.
Last September six of his siblings all appeared in TV adverts asking his constituents in Arizona to vote for his Democratic opponent.