The millionaire behind India’s answer to Starbucks has disappeared, triggering a search, suspicions of suicide and a 20% plunge in the shares of his company, Coffee Day Enterprises Ltd.

V.G. Siddhartha is chairman and managing director of the company behind India’s biggest coffee chain, Café Coffee Day, which has more than 1,700 outlets compared with the 150 stores Starbucks has in India. He was last seen Monday evening on a bridge in the city of Mangalore, said Hanumantharaya, deputy commissioner of police in the southern...