In most developing nations, the collapse of Satyam Computer Services would follow a well-worn arc: poor corporate governance in a poorly regulated land leads to criticism from the West and promises of swift action from the government.

But Satyam’s collapse happened in India—and has left that nation, increasingly vital to the supply chain of U.S. companies, flummoxed about what it should do.

Corporate governance among large businesses in India is generally considered good, because so many companies there want to do business with the West. Satyam in particular boasted a board of esteemed business experts, listed its shares in the United States, ...