He invited Mr Bandt to "end the problem" and avoid legal action by making a public apology.
"I hereby apologise for those statements," Mr Bandt said in a statement on Thursday afternoon.
Senator Molan said he was deeply disappointed in the apology.
"Really there seemed to me to be no contrition that I think was proportional to the serious allegations that he made," Senator Molan told 2GB radio.
"I want him to reconsider it, and I consider that all my options are still on the table, so I'm not accepting it."
Home Affairs Minister Peter Dutton described Senator Molan as a "hero" who'd put his life on the line while people like Mr Bandt were "back home burning flags".
"He wouldn't have the guts or the capacity or the ability to serve his country like Jim Molan did," Mr Dutton told 2GB radio.
"Adam Bandt is a horrible individual. He demonstrates it on a daily basis in parliament and frankly this just exemplifies it again in his pathetic comments about Jim Molan this week."
Senator Molan attracted widespread criticism after re-posting anti-Islam videos by far-right group Britain First on his Facebook page last year.
He admitted he was "unwise" to post them.
Greens Leader Richard Di Natale would not back down over his decision to raise questions about Senator Molan's military record in Iraq during Question Time in the Senate.
"No. I don't think we have anything to apologise for," Senator Di Natale told ABC TV.
"The reality here is there is a culture of silence that has emerged between the Liberal Party and the Labor Party that says we should never question the activities of the Defence Force."
Senate crossbencher Derryn Hinch said the Greens were acting "foolish and cowardly" for taking on someone from the armed forces.
"To be called a war criminal when you've proudly represented your country and none of those people sitting next to me in that parliament, I doubt, have ever faced a bullet," he told ABC TV.