NEW DELHI — Devyani Khobragade, the Indian diplomat accused of visa fraud for allegedly underpaying her babysitter, left the United States after she was indicted in a case that roiled relations between the two countries.

Khobragade, 39, was charged Thursday with making “multiple false representations” to U.S. authorities to obtain a visa for the caretaker, and the State Department later ordered her to leave the country after India denied waiving her diplomatic immunity.

Her flight has already left the United States, according to an Indian government official who asked not to be identified because he was not authorized to speak publicly.

Khobragade’s departure may resolve a diplomatic row that threatened to jeopardize a growing economic relationship as annual trade in goods and services between the countries nears $100 billion.

The dispute, which erupted after reports that she was strip-searched, put a cloud over President Barack Obama’s goal of strengthening U.S.-India ties.

“Even though the case is being resolved, it caused waves that will take time to resolve and it has caused reputational harm on both sides,” P.J. Crowley, a former State Department spokesman, said in an email. “Both sides need to make conciliatory gestures in the aftermath,” he wrote. “Wounds do heal, but as always, they leave a scar.”

India’s Ministry of External Affairs said Friday Khobragade had been transferred to a post in New Delhi and that it had declined a U.S. request to waive her diplomatic immunity. Syed Akbaruddin, a ministry spokesman, didn’t answer calls to his mobile phone.

Khobragade was first charged Dec. 12. Her case triggered Indian outrage when news circulated that she was arrested by the State Department’s Diplomatic Security Service in front of her daughter’s school in upper Manhattan and strip-searched while being held with other female suspects. She was released on $250,000 bond which was unsecured.

Manhattan U.S. Attorney Preet Bharara has said the strip- search was standard practice in an arrest. The incident sparked an uproar in India as the nation of 1.2 billion people prepares for elections in a few months.

“India will continue to be upset about how this was handled and rightfully so, but this has opened the door to putting this issue behind us,” said Karl Inderfurth, a former State Department official responsible for South Asia.

Secretary of State John Kerry should now reach out to the Indian government, Inderfurth, now a senior adviser at the Center for Strategic and International Studies in Washington, said in an interview.

“I’d hope the last word is from the State Department, preferably from Secretary Kerry, stating U.S. regret, if not apology, for how this was handled,” he said.

The Indian government demonstrated its displeasure with a variety of small reprisals, including removing some security barriers at the U.S. Embassy in New Delhi, lifting traffic-violation exemptions for U.S. Embassy cars, and ordering the American Center, a venue in central New Delhi for U.S. cultural programs, to halt its activities.

The U.S. countered by postponing anticipated visits to India by Energy Secretary Ernest Moniz and Nisha Desai Biswal, the U.S. assistant secretary of state responsible for India.

During a visit in November 2010, Obama called the relationship with India “one of the defining and indispensable partnerships of the 21st century.”

Indian Prime Minister Manmohan Singh, who was Obama’s first official diplomatic guest in 2009, last week described a deal with the United States that allowed it to import nuclear technology as his greatest achievement during a decade in power.

He told reporters in New Delhi that diplomacy should be given a chance to resolve the recent “hiccups” in relations.

Raymond Vickery, a top U.S. trade official under former President Bill Clinton and now a senior director at the Albright Stonebridge Group in Washington, said that expanding trade and investment has been “the underlying driver” in U.S.-India relations.

Speaking before the late legal developments, Vickery said business relations hadn’t yet been hurt by the dispute. The bilateral trade in goods and services reached $92.5 billion in 2012 from $59.9 billion in 2009, the first year of the Obama administration, according to a joint statement following U.S.-India economic talks in October.

Indian foreign direct investment in the United States increased from $227 million in 2002 to almost $5.2 billion in 2012, making India one of the fastest growing sources of investment into the U.S., according to the statement.

At the hearing in Manhattan, Khobragade’s lawyer Daniel Arshack said he told her not to board an Air India flight Thursday afternoon because “there was at least a possibility that it would be viewed as flight” from prosecution.

Arshack said his client had “diplomatic status” and told U.S. District Judge Shira Scheindlin that she no longer had jurisdiction over the case.

“I was not willing to permit her to depart without appearing before your honor,” Arshack said. He asked Scheindlin to vacate his client’s bail.

Scheindlin later agreed that Khobragade wouldn’t be accused of bail jumping if she agreed to the State Department’s request. The judge deferred a decision on bail to a later date.

“We are pleased that the United States Department of State did the right thing today by recognizing the diplomatic status to which Dr. Khobragade has always been entitled,” Arshack said in a statement issued last night. He accused the government of committing a series of “blunders.” 

The visa fraud charge against Khobragade carries a maximum prison term of 10 years, while the false statements charge has a maximum term of five years, according to prosecutors in Bharara’s office.

In a contract Khobragade submitted as part of the visa application, the diplomat said she paid the babysitter $9.75 an hour — above minimum wage as required by law, State Department Special Agent Mark Smith said in the original criminal complaint.

In a second contract, the diplomat agreed to pay the babysitter 30,000 rupees a month, or $573, the United States said, which came out to $3.31 an hour. New York minimum wage is $7.25 per hour.

“Khobragade did not want to pay the victim the required wages under U.S. law or provide the victim with other protections against exploitative work conditions mandated by U.S. law [and widely publicized to foreign diplomats and foreign officials],” according to the indictment.

After her arrest, Khobragade was named by her country to serve as a member of its permanent mission to the United Nations, a position that gave her a higher level of diplomatic immunity than she enjoyed as deputy consul general at India’s consulate general in New York.

The United States accepted the request to accredit Khobragade to the U.N. mission, according to a State Department news release. Seeking to deny such a request would be almost without precedent, except in matters of national security including espionage, the department said.

The United States first asked for a waiver of diplomatic immunity, which India subsequently denied, then requested that she return to India, the department said.

The case is U.S. v. Khobragade, 14-cr-00008, U.S. District Court, Southern District of New York (Manhattan).