I agree with most of what the others have said.
BUT I think she has a point if she was just objecting to your putting family information online, rather than in your doing the research per se. I am one of many who would never put a tree on Ancestry or Genes. No website is really secure. You have no idea where the information will end up. I can find a large tree on Ancestry which contains ALL the names in your profile (as well as many others); it is probably not yours, as you say it is private. I just say this to illustrate my point.
Some of my relatives are against family history research for similar reasons - my grandmother was a very private person and would have hated the idea of her picture on the internet. With tact, persuasion and showing them interesting things, some have come round. Doing some courses (in research methods and recently oral history, so I can do "proper interviews") helped as well, as it gave my research a stamp of authority.
But as for her general objection to your research, as others have said, there is nothing she can do to stop you. She may well come round if you employ your best diplomatic skills, slowly over the years. But perhaps reconsidering having your tree online might win her over eventually!