Kenny Jackett resigned as Millers boss following the home game with Leeds United last weekend and Paul Warne has been placed in caretaker charge ahead of Saturday's trip to Burton.
Rotherham are now looking for their sixth manager in just 14 months and Stewart says they will not rush the appointment.
"I am an ambitious person that will not change, just because we have some bad days doesn't mean we'll quit," he said on the club's website.
"We are not quitters at Rotherham United, we are fighters.
"We have seen what can be achieved here in the past. There is no reason why that cannot be achieved again. That’s what we are looking for.
"When we appoint managers, we appoint them in good faith because we feel that they are the best candidate to take Rotherham United forward.
"In the past we genuinely believe we have got it right with managers, on occasions it hasn’t worked out. That’s not just Rotherham United; it can be said of every football club."
He added: "What we need to remember is that we are still a Championship Club, we have come a long way over the past few years and, whilst I believe we should be doing better, we remain proud of what we have achieved at this Club.
"The Club as a whole are fully committed to doing the very best for Rotherham United. That will always be the case."