"We are shocked and appalled at what the US did. India is forcefully taking up with the US the treatment of the arrested diplomat. It is not acceptable," the foreign ministry said.
Ms Khobragade, Deputy Consul General at the Indian Consulate in New York, has been accused of lying on the visa application for an Indian national who worked for her for less than four dollars an hour.
She was reportedly arrested on the street while she was dropping her daughter to school. She was later released on a $250,000 bail.
She was arrested on the orders of Indian-born Preet Bharara, Manhattan's top federal prosecutor, who said Ms Khobragade had presented false documents for the woman who worked as a babysitter and housekeeper at her home from November to June, and lied about her wages.
He alleged the woman, Sangeeta Richard, had been exploited and made to work for "less than fair" wages. "This type of fraud on the United States and exploitation of an individual will not be tolerated," he said.
Ms Khobragade, 39, who has two young daughters, is in charge of Political, Economic, Commercial and Women's Affairs at the consulate. The charges against her carry a maximum sentence of 15 years in prison.
India has asked the US to resolve the matter with "sensitivity", taking into account Ms Khobragade's diplomatic status. In a statement, the Indian embassy in Washington said Sangeeta Richard had been "absconding since June" and faces an arrest warrant for cheating and extortion. (Read: Full text of Indian Embassy's statement)
Sangeeta Richard has alleged that she was to be paid $ 4,500 per month according to the employment contract furnished for her visa, but Ms Khobragade later changed it to $ 537 a month. She also complained that she was forced to work more than 40 hours a week.
Ms Khobragade's lawyers argue that with her own salary of $ 4,000 a month, she can hardly pay her help more than that. They also said her arrest on the street, without being given a chance to explain, was "shocking and unprecedented."
Daniel Arshack, Ms Khobragade's lawyer, told NDTV "If what you are referring to is that she was arrested on the street while dropping her daughter off at school, I think there is no precedent for that and hopefully there never again will be such a precedent. It is shocking to me that she was not provided with an opportunity to do what any responsible person would do, that is, if they wanted to talk to her she would have come with her attorney and spoken to them."