The two countries cooperate on a wide range of issues including counter-terrorism, regional security and defense. India is also a major market for U.S. weapons.
Khobragade returned to New Delhi on Friday following a deal that let her leave the United States after being indicted by a federal grand jury.
Since her return she has limited comment on the case to expressing thanks for the support she has received in India, where her treatment caused fury in the foreign service and broader antagonism towards the United States.
"I'm really grateful for all your support, my government will speak for me, my lawyer will speak for me," she told reporters on Saturday.
Call for closer cooperation
Khurshid said Washington should have asked India to withdraw Khobragade rather than arrest her.
"When you have two friendly countries when something is becoming untenable, then you cooperate with each other," he said.
"If you have inkling or a sense that something like this will unfold, you should take the quickest steps that you can to ensure that it doesn't happen."
India sharply curbed privileges offered to U.S. diplomats in retaliation for the treatment of Khobragade. Khurshid gave no sign these measures would be reversed.
(Read more: India tells US to close embassy club)
India on Friday asked Washington to withdraw a diplomat from New Delhi, in response to what was effectively the expulsion of Khobragade from the United
States. An Indian official told reporters the government believed the U.S. diplomat had a role in the Khobragade case but gave no more details.
Washington has agreed to withdraw the envoy, who it did not identify. Citing sources, Indian media said the diplomat was a senior security official at the
U.S. embassy who India believes helped the housekeeper Sangeeta Richard's husband flee to the United States despite a court injunction against him in Delhi.
The continued presence in the United States of the housekeeper could pose a challenge to Washington as it seeks to repair its relationship with New Delhi.
India has asked U.S. authorities to arrest Richard over the Indian government's allegations that she stole cash, a mobile phone and documents from Khobragade. Richard has denied the charges.