He was not just my brother, he was my best friend. He will be greatly missed, but we will meet again,” Mr Trump said.
“It is with heavy heart I share that my wonderful brother, Robert, peacefully passed away tonight,” Mr Trump said in a statement.
“He was not just my brother, he was my best friend. He will be greatly missed, but we will meet again.
“His memory will live on in my heart forever. Robert, I love you. Rest in peace.”
The president visited his younger brother in hospital on Friday, after it was revealed he had been “very ill”.
However, Robert failed in several court actions to stop publication of the book, which is called Too Much And Never Enough: How My Family Created The World’s Most Dangerous Man.
Robert argued that Mary Trump had violated a non-disclosure agreement she signed with him, the president and their sister, retired federal appeals court judge Maryanne Trump Barry.
Mary Trump and her brother allegedly signed the agreement with their uncles and aunt over their share of the estate of Fred Trump Senior, the president and Robert Trump’s father, and Mary’s grandfather.
Mary’s father, the late Fred Trump Junior, died in 1981.
Despite their different personalities, the president was said to be very close to Robert, calling him “much quieter and easy-going than I am” and “the only guy in my life whom I ever call ‘honey’.”
Robert had run some of Mr Trump’s projects in the 1980s, and managed his family’s real estate portfolio. He was also part of New York’s social scene.
However, during his brother’s presidency, Robert stepped away from the limelight and retired to the Hudson Valley.
He said he supported his brother’s White House run “1,000%” during an interview in 2016.