Fauci is a Federal employee, so not sure that he could just be fired.
The President does, however, have the power to sideline Fauci, keeping him away from press briefings and media interviews — as has happened in recent weeks, though Fauci has proven adept at pushing his message through different channels.
CNN spoke with Max Stier, president and CEO of Partnership for Public Service, about the history of the federal government’s protections for civil servants and what might happen if Trump tried to go ahead with such a move. The conversation, conducted over the phone and lightly edited for flow, is below.
CNN: Tension between the President and Fauci has escalated in recent weeks and days. Could Trump fire Fauci and what would that look like?
MS: The President could not fire Fauci without cause. There are civil service protections for career federal employees that prevent them from being removed or demoted for political reasons.
https://www.cnn.com/2020/07/13/politics/can-trump-fire-fauci/index.html
So basically I was correct in my assumption, so it sounds from what I read.