NEW DELHI—McDonald’s Corp. has a baffling problem in India. It can’t get restaurants there to stop selling its food.

In August, after a lengthy dispute, the fast-food giant ordered its partner of 22 years in northern and eastern India to stop using the McDonald’s brand and system. But Connaught Plaza Restaurants Pvt. Ltd., which operated 40% of McDonald’s Indian outlets, about 150 restaurants, went rogue.

It...