The nuances of that dispute, however, have gotten lost in the increasingly hostile exchanges between the two men. In July, after Mr. Paul accused him of lying to Congress, Dr. Fauci shot back, “If anybody is lying here, senator, it is you.” Last month, Dr. Fauci arrived at a Senate hearing brandishing a fund-raising webpage for Mr. Paul that included a “Fire Dr. Fauci” graphic, and accused Mr. Paul of exploiting the pandemic for political gain.Later in that same hearing, Dr. Fauci muttered under his breath that Mr. Marshall was “a moron” — a comment caught on an open microphone — after the senator posted Dr. Fauci’s salary on a placard and demanded his financial disclosure forms, suggesting he might be engaged in financial “shenanigans” with the pharmaceutical industry.(Dr. Fauci’s financial disclosure forms, which Mr. Marshall has since posted on the internet, show investments in bonds and mutual funds, not drug companies. He is paid an annual salary of $434,312 under a provision that allows government doctors and scientists to be highly compensated, akin to what they could earn in the private sector.)Dr. Fauci said he did not regret the “moron” remark, or the pushback against Mr. Paul. But Ms. Timken said calling Mr. Marshall a moron was “beyond the pale.”Even some Fauci fans in academia and government say he might have been better off keeping his cool to avoid amplifying his Republican critics and alienating voters who need to hear his public health message. Some suggest he lower his profile; he says the White House asks him to go on TV.“He’s been pushing back in a way that is not common for us to see for American scientists, and I don’t know if it’s a good idea,” said Dr. Ashish K. Jha, the dean of the Brown University School of Public Health.If Democrats lose seats in the midterm elections, as many expect, Dr. Fauci may have a Republican-controlled Congress to contend with. Another Republican from Ohio, Representative Jim Jordan, who claims that Dr. Fauci knew the coronavirus “came from a lab,” has vowed that Republicans will investigate him if they win control of the House.