I don't know if you need consent to take your granddaughter to the USA. If she has her own passport, I suppose she could travel to the US with or without you. To be certain, you can contact the US Embassy in your country and pose this question.
If you are traveling to the USA with your grandaughter and she is under 18 years of age, I suggest that you obtain a temporary guardianship of her for the duration of the trip. If she should be involved in an accicent that requires medical attention, the US hospital or the physician may require written permission from the child's parent or guardian before they will perform certain medical procedures on her. Even some of the more adventuresome tourist attractions, require written permission of a child's parent or guardian before they will allow the child to participate.