The letter outlines a 30-day deadline for the body to commit to "substantive improvements" or risk losing millions and US membership altogether.
Addressed to WHO chief Tedros Adhanom Ghebreyesus, it criticises stages of the body's response since December.
Earlier on Monday, Mr Trump called the UN's health body a "puppet of China".
The president, who faces re-election this year and has himself been criticised for his handling of the pandemic, has blamed China for trying to cover up the outbreak and has accused the WHO of failing to hold Beijing to account.
China has repeatedly denied the allegations, arguing the country had released information "in an open, transparent, responsible and timely manner".
The US has more than 1.5 million of the world's 4.8 million confirmed cases of coronavirus so far, with more than 90,000 deaths.