PCS calls for an end to the "bullying culture" that the government has presided over in the civil service.
Home secretary Priti Patel is coming under mounting pressure over allegations of bullying following the resignation of Home Office permanent secretary Philip Rutnam last weekend.
Mr Rutnam has said he will take a claim of constructive dismisal against Ms Patel and the government.
Media reports have suggested the home secretary is facing multiple accusations of bullying both from her time as home secretary and while she was at the Department for Work and Pensions and Department for International Development.
Responding to the fallout, PCS General Secretary Mark Serwotka said: "This government has presided over a bullying culture within the civil service which they have failed to tackle.
"So it's a bit rich of Priti Patel to talk about coming together when questions around her role in bullying claims remain unanswered.
"There should be an independent inquiry into bullying in the civil service.
"Our union along with others have been working with the cabinet office arguing for a clear procedure on handling complaints of bullying, discrimination and harassment, particularly regarding allegations against ministers MPs and Special Advisers.
"Despite assurances that this would become part of the ministerial code, this appears to have been abandoned."