MUMBAI—India’s liquor market was for years a thorn in the side of Diageo PLC, the world’s largest spirits maker. Then it met the “King of Good Times.”

Vijay Mallya, a billionaire fond of mullet haircuts, diamond earrings, sports and lavish parties, invited Diageo executives to his Mumbai residence in early 2012 and offered them a deal for a majority stake in India’s largest liquor maker, which he controlled.

Diageo...