Wakefield Council is calling for Prime Minister Theresa May to sack the Secretary of State for Health and Social Care.
Councillors, on Wednesday, voted in favour of a motion calling for Jeremy Hunt to lose his job, after an emotional discussion about the country's NHS.
The authority is also calling for "adequate" funding for the health service and an end to its "chronic neglect".
The motion, set out by Coun Steve Tulley, said the NHS was under "severe pressure" due to "continued under-funding and neglect by the Conservative government, along with the worst winter crisis on record", putting increasing demands on staff.
It said locally there was a 6.4 per cent increase in A&E attendance in December compared to the previous year and noted that the average bed occupancy at Mid Yorkshire Hospitals NHS Trust, which runs Pinderfields, Pontefract and Dewsbury hospitals, was 96.55 per cent from November 27 to January 7.
Introducing the motion, Coun Tulley said: "I would ask this council today to call on the Prime Minister first of all to get shut of an inept, useless Secretary of State for Health and Social Care and start looking at funding. The money is available"
The discussion that followed saw several councillors share their own experiences of the NHS, as well as those of friends and family members.
In the public gallery, signs saying 'save our NHS' and 'save our A&E' had been stood on display.
Coun Pat Garbutt said: "The government need to act now. Our citizens deserve the very best care and the staff deserve to be supported in the vital work they do."
Coun Nadeem Ahmed, leader of the opposition party Wakefield Conservatives, said he could agree with "98 per cent of the motion" but said sacking Mr Hunt would not solve any issues.
He said: "The government has pledged more money but I don't think it is enough. I think there's a wider question to ask about the way that the NHS is funded and what we needto do to move it forward."
He suggested taxation could be increased to increase the funding available.
The motion said the council recognised hospital trusts are doing "everything possible" to provide the best care for patients and commended staff for their hard work and dedication "under extremely difficult circumstances".
It was carried after all Labour councillors present and one independent voted in favour. Five Conservative councillors voted against the motion and there were two abstentions.