President Donald Trump’s push to impose painful tariffs on auto imports has put close U.S. allies in the crosshairs of a global trade row that is riling stock markets and creating uncertainty among auto makers, investors and governments.

Shares of some of the biggest international auto makers including Volkswagen AG, Toyota Motor Corp., BMW AG and Daimler AG, which have big exposure to the U.S., fell Thursday, a day after the U.S. Commerce Department launched a probe into whether it could raise tariffs to up to 25% on auto...