The Sixth Sense actor was detained by cops at a ski lodge in Mammoth Lakes, California, on April 8 and has since been charged with possession of cocaine and disorderly conduct.
He has now offered a public apology after bodycam footage showed him calling a police officer a "f*****g Nazi" and using an anti-Semitic slur.
The star - who lost his home in the California wildfires earlier this year - said in a statement given to PEOPLE: "I'm absolutely horrified by my behaviour. Had I known I used this disgraceful language in the throes of a blackout, I would have spoken up sooner.
"The past few months of loss and displacement have broken me down to a very low emotional place. But that's no excuse for using this disgusting word.
"From the bottom of my heart, I apologise to absolutely everyone that this hurts.
"What came out of my mouth was nonsensical garbage. I've let the Jewish community down and it devastates me. I don't ask for anyone's forgiveness, but I promise to atone for my terrible mistake."
The Mono County District Attorney's office has confirmed Osment was charged with two misdemeanour offensives - disorderly conduct under the influence of alcohol in public and possession of cocaine - and is due to be arraigned at Mono County Superior Court on July 7.
Police are believed to have arrived on the scene of the alleged incident after they received a report of "unruly conduct" by the actor.
The bodycam footage seen by PEOPLE is believed to show a witness telling a police officer that Osment had been drinking in a nearby bar, but staff there stopped served him after he spilled a drink.
He is alleged to have become "aggravated" and "argumentative" and attempted to get on a ski lift, but employees didn't let him and called in police.
In California, disorderly conduct is punishable by up to six months in jail and a $1000 fine, while a first offence for possession of cocaine can lead to community service or a fine of at least $1000.
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