KYIV, Ukraine—Ukrainian President Volodymyr Zelensky is withholding his support for a proposed parliamentary investigation into a gas company entangled in the U.S. impeachment inquiry.

A dozen members of Ukraine’s parliament this fall decided to seek an investigation by the legislative body of Burisma Holdings, the gas company that in 2014 put then-Vice President Joe Biden’s son Hunter on its board. The lawmakers—chiefly from opposition parties but including some members of Mr. Zelensky’s party—submitted a request to the president...