On the final night of his visit to Washington in late June, after 15 standing ovations in Congress and an opulent White House dinner tailored to his vegetarian tastes, Prime Minister Narendra Modi of India set time aside to court and be cheered by another important constituency: the Indian diaspora.

Backstage at the Kennedy Center, as business leaders in bespoke suits and fine silk saris filtered into a 1,200-seat theater, Mr. Modi met with a handful of entrepreneurs. Most were young, educated in India, made rich in America, and eager to connect with the man who presents himself as a guru to the world, preaching how this is “the century of India.”

“Thank you for lifting the image and spirits of Indian Americans,” Umesh Sachdev, 37, told the prime minister, explaining that he was the founder of Uniphore, an artificial intelligence business