You should have not checked your daughter and needed to arrange another good gynecologist for her problem hence you done mistake there.
Second things you should have kept your wife with you when you were examining your daughter but you failed to do it.
Third, you after examining her, you should have not applied ointment on her genital and could have arranged another staff from your clinic to help you.
Another blunder you did was, she asked you that she felt itching deep inside in her vagina, being a doctor you could easily caught her lying but you purposely did not do anything and followed her order like lover.
Sorry doctor you are keep talking about on moral issue but I find you as an immoral and sex starved man. You already crossed the entire limit, which you should have not crossed as a father.
Now she is pressuring you to have sex with you otherwise she will expose your devil face to your wife and that would be disaster for you I am sure so you have no option than to follow her order and start having sex with her.
She is young and you are looking for a vagina so do not lose such opportunity to have sex as I do not think you can find a such young girl for sex.
I am suggesting you to go ahead because your question sounds that you want to get into her pants but you should have to keep moral and principles aside, as morally, it is wrong to have sex with daughter and no arguments can justify your act with your daughter.
However, if you really do not want to fuck your daughter then act as a father and warn her with consequences. Also, clearly tell her that you are not afraid of her threats. I do not think that she will go and disclose everything to your wife. You have to take risk if you want to play the role of responsible father.