Kennedy, commonly known by his initials RFK Jr, has a history of spreading health information that scientists say is false.
If his nomination is ratified by the Senate, he will lead a huge agency overseeing everything from food safety to medical research and welfare programmes.
The announcement came amid a fresh flurry of nominations on Thursday evening, with Trump also declaring his intention to nominate North Dakota Governor Doug Burgum as his interior secretary.
Trump said he would formally announce the selection of Burgum - a former businessman who ran against the president-elect for the Republican presidential nomination - on Friday.
He had initially teased the move during a speech to supporters at Mar-a-Lago - his first since election night - saying he would be appointing Burgum to a "very big position" - before seemingly deciding to dispense with the suspense.
Other nominations announced on Thursday include:
- Former Georgia Congressman Doug Collins as secretary of veterans' affairs.
Todd Blanche, Trump's defence lawyer in his "hush money" criminal trial, to serve as deputy attorney general.
- Dean John Sauer, who represented Trump in a US Supreme Court case earlier this year, as solicitor general. He will be charged with supervising and conducting government litigation in the Supreme Court.
- Jay Clayton, former chairperson of the US Securities and Exchange Commission, as US attorney for the Southern District of New York, one of the most influential federal trial courts.
Trump said in a statement he was "thrilled to announce" Kennedy's nomination.
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