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 a statement announcing his decision.
Nominated by president Ronald Reagan, the 81-year-old Kennedy took his oath of office in 1988.
He said his decision to step aside was based on his deep desire 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 would be selected from among a list of 25 possible candidates -- 20 identified during his presidential campaign, and five added since.
"We have a very excellent list of great, talented, highly intelligent, hopefully tremendous people," he said.
In a statement the White House said that "Justice Kennedy has been a tireless voice for individual rights and the Founders' enduring vision of limited government. His words have left an indelible mark not only on this generation, but on the fabric of American history."
Kennedy's vote, positioned at the very center of the court, gave him a place of special prominence among the nine justices.
He has cast the deciding vote in a number of historic cases, including the high court's groundbreaking decision in 2015 legalizing gay marriage, a five-to-four decision for which he wrote the opinion.