European Competition Commissioner Margrethe Vestager has become a household name in Silicon Valley since she orderedApple Inc. to pay $14.5 billion in alleged unpaid taxes to Ireland and finedAlphabet Inc.’s Google a total of $7.8 billion for allegedly abusing its dominance to clobber rivals.

In June, President Trump referred to her as the EU’s “tax lady” and said she “really hates the U.S.”

Ms....