Is it considered a grave sin for a woman to go to a male doctor for treatment with cauterization while he views her private parts, and is expiation obligatory upon her, or is repentance sufficient?

You must repent to Allah Almighty for revealing your private parts to a doctor without necessity, for this act violates the Sharia and breaches modesty, and it has no expiation except repentance. You should keep this matter secret and not inform anyone about it, especially your husband.