President Donald Trump’s nominee for the top attorney job at the NLRB argued against a now-defunct air traffic controllers union in a case that critics consider a key event contributing to decades of decline in union membership and wage stagnation.

Peter Robb was nominated Sept. 15 by the White House to serve as general counsel of the National Labor Relations Board. Robb was the lead attorney in a controversial case that resulted in the firing of thousands of striking workers...