First, go to jail. He must be seen as paying a real price for his most recent bad behavior.
Second, offer to give all the money he makes in the next season of his show to charity – maybe spread it around a bit and include a bunch of good causes from supporting victims of violence and homeless people to research into substance abuse.
Third, release a well crafted statement of apology. Vet it first through a focus group that can attest to it sounding sincere.
Fourth. Have your father, the much respected Martin Sheen, come out and make a few comments about how you obviously have real problems that have needed addressing for a long time.
Fifth. Clean up the mess of bad old comments online with a concerted content replacement campaign.
Sixth. If none of this has much impact, offer to cut off your pinky finger in an act of Yakuza-like contrition. Publicize you are going to do it and, unless your kids beg you not to, follow through with the plan.