Kennedy's departure at the end of July will give Trump the opportunity to appoint his second justice to the nine-seat bench, which exerts a deep and far-reaching influence on American life.
"It has been the greatest honor and privilege to serve our nation in the federal judiciary for 43 years, 30 of those on the Supreme Court," Kennedy was quoted as saying in announcing his decision.
Nominated by president Ronald Reagan, the 81-year-old Kennedy took his oath of office in 1988. He said he was stepping aside to spend more time with his family.
Paying tribute to Kennedy as "a great justice of the Supreme Court," Trump told reporters the replacement process will "begin immediately."
"Hopefully we will pick someone who is just as outstanding," he told reporters in the Oval Office.
He said Kennedy's successor - who like all Supreme Court justices will be appointed for life - would be selected from among a list of 25 possible candidates, 20 identified during his presidential campaign, and five added since.