An inspector general’s finding that a top Donald Trump labor appointee should have recused himself from a major pro-employer ruling has triggered Democratic calls for hearings, a demand for answers by two U.S. senators, and news that a separate IG report is on the way over whether the appointee committed misconduct.

On Feb. 15, the chair of the National Labor Relations Board disclosed to Congress a report from the agency’s inspector general, David Berry, concluding there was “a serious and...