The president cannot directly fire Dr. Anthony Fauci because he is not a presidential appointee.
Technically, the president cannot directly fire Dr. Anthony Fauci because he is not a political appointee. As a career federal employee and director of the National Institute of Allergy and Infectious Diseases since 1984, Fauci is protected by federal civil service regulations that shield him from being fired or demoted for political reasons. Fauci could be removed, but it would consist of a complicated process layered with civil service protections that require the government agency to provide evidence that there is a just cause for dismissal, including failure to follow orders or misconduct. The process to remove him would need to be initiated by someone in Fauci’s chain of command, such as the director of the National Institutes of Health or the health and human services secretary.
However, while these rules are still in place, a controversial executive order issued by Trump could remove these long-held service protections from tens of thousands of civil servants, making it easier to dismiss them with little cause or recourse, but Trump hasn't take any action yet.