Theresa May has defended Mr Hunt, describing him as an “excellent Health Secretary”.
But Mr Smith, the Labour leadership hopeful, believes the Prime Minister should fire Mr Hunt who he claims is the “worst Health Secretary in the NHS’s history”.
He said: “It is simply astounding that on the day Theresa May walked into Number 10 one of the first decisions she made was to keep Jeremy Hunt in post.
“It highlights the weakness of the Labour Party under Jeremy Corbyn that Mrs May felt she could keep him on in the job, despite him having lost the faith and trust of patients and staff.
“This loss of trust has been brought about in no small part by his shameful misuse of statistics for political ends.
“We need to find a way to put the NHS back on the road to recovery and avert this strike, it is clear the Health Secretary is unable to do so.
“That is why I am calling on Jeremy Hunt to be sacked immediately.”
Mr Smith said if Mrs May does not sack Mr Hunt, Labour should put a vote of no confidence in the Health Secretary when Parliament returns on Monday.
The strike action is due to take place from September 12-16 with more walkouts planned in the future.