The apology offered privately cannot fully correct an accusation made publicly.

Harm done in front of people leaves something behind in those people, and a private apology cannot reach it. The person you accused walks back into the same room carrying what everyone there heard, while your regret stays in a conversation only the two of you attended. Sincerity is not the missing part. Reach is.

If you have said something about someone in front of others, ask them what would help, and be willing for the answer to be a public correction. Say it plainly and without a second round of explanation. The repair belongs in roughly the same place the damage landed.