There are no formal federal rules that prohibit the younger Mr. Trump from pursuing his business interests; he is a private citizen. And even though President Trump is still a beneficial owner of the family business, he is not subject to conflict of interest laws, as the president is exempt. The White House, asked if Donald Trump Jr.’s trip gave even an appearance of a conflict of interest, declined to comment on the matter.

India is the Trump Organization’s biggest international market, with four real estate projects underway. The largest is here in Gurgaon, a fast-growing city outside New Delhi with a litany of Fortune 500 companies and a skyline filled with skyscrapers. The Trump development encompasses two towers with a total of 254 “ultra luxury” units, according to the company, each selling for up to $1.5 million.

The luxury apartment buildings near the Trump development in Gurgaon have names like Palm Springs and Central Park, even as they overlook pothole-ridden highways. The Trumps and their partners are offering buyers the opportunity to become “members of the Trump family,” with the promise of exclusive amenities such as an infinity swimming pool, a billiards room and valet services. The marketing materials, however, don’t mention the president — his name and image, as well as that of his daughter Ivanka, who joined his administration, were removed from brochures, websites and billboards from projects in India and elsewhere in the world.

Still, confusion among Indians is evident.

“I had no idea that Trump built property around the world before this development was launched,” said Hitesh Khanna, who is helping sell units at the Trump towers in Gurgaon through a local real estate brokerage firm. “I just thought he was president.”