Ask HN: Accused of acting sexist. What do do?

A woman between 20 and 30 has accused me of mistreating her because of her gender. She says that I have asked her questions I wouldn't have asked any man. I don't think that is true because I have asked almost word-for-word identical questions to male colleagues. And they have all been work-related. Like "Any news from project XYZ?" Nothing about her personal life. I have told her that if I have offended her I apologize and that it wasn't my intention. She said that she doesn't believe me and that I intentionally denigrate her. Since then I have avoided her because I fear that she might accuse me of more and worse things. But we work in the same project so it isn't always possible. I'm afraid that false rumors will spread about me. She has many friends in the department I have none. I also have Asperger's and have been accused of taking things literally in the past.