BERLIN—Volkswagen AG’s crisis over alleged manipulation of U.S. emissions tests threatened to engulf its global business as European regulators put the German auto maker in their cross hairs and the U.S. launched a criminal probe into the matter.

Its shares plunged as much as 20% on Monday in the wake of the U.S. Environmental Protection Agency’s disclosure last week that the company could face fines of as much as $18 billion for allegedly manipulating exhaust-emissions tests. Volkswagen acknowledged that it installed software...