A wide-reaching labor rights protections bill is just a week old, but it’s already racking up vociferous opposition from business groups and Republicans.

The Protecting the Right to Organize Act (H.R. 2474) introduced in the House and Senate May 2, would substantially broaden workers’ right to strike and unionize and would implement more tools for the National Labor Relations Board to punish employers for blocking a union campaign. The measure also would expand the definition of employee in an effort...