President Barack Obama said he will nominate a successor to U.S. Justice Antonin Scalia "in due time."

Scalia died Saturday of natural causes, leaving a vacancy on the bench of the U.S. Supreme Court. The president said today is a time to remember the justice's nearly 30-year legacy – not hammer out the finer details of how and when to replace him.

"I plan to fulfill my constitutional responsibilities to nominate a successor in due time," Obama said in prepared remarks from Rancho Mirage, California, where he spent the afternoon golfing. "There will be plenty of time for me to do so, and for the Senate to fulfill its responsibility to give that person a fair hearing and a timely vote."

GettyImages-510097616 A group of people pray in front of the U.S. Supreme Court, Feb. 13, 2016 in Washington, D.C. Supreme Court Justice Antonin Scalia was at a Texas Ranch Saturday morning when he died at the age of 79. Photo: Drew Angerer/Getty Images

Scalia's death set off a firestorm of debate over who should replace the 79-year-old justice in the heat of a presidential election cycle. Scalia was a staunch conservative and vocal opponent of the Obama's landmark policies, including the Affordable Care Act, the Clean Power Plan to curb power plant emissions and immigration reform.

Several Republican presidential hopefuls and Senate Majority Leader Mitch McConnell, R-Kentucky, said the next president – not Obama – should elect Scalia's replacement.

"The American people should have a voice in the selection of their next Supreme Court Justice," McConnell said in a statement Saturday. "Therefore, this vacancy should not be filled until we have a new President."

House Judiciary Committee Chair Chuck Grassley, R-Iowa, similarly remarked that, "The fact of the matter is that it’s been standard practice over the last 80 years to not confirm Supreme Court nominees during a presidential election year."

Hillary Clinton rebuffed calls to delay replacing Scalia. "The Republicans in the Senate and on the campaign trail who are calling for Justice Scalia's seat to remain vacant dishonor our Constitution," the Democratic presidential candidate and former U.S. secretary of state said in a statement. " The Senate has a constitutional responsibility here that it cannot abdicate for partisan political reasons."

GettyImages-510093578 U.S. President Barack Obama speaks on the death of Supreme Court justice Antonin Scalia in Rancho Mirage, California, Feb. 13, 2016. Photo: Mandel Ngan/AFP/Getty Images

Obama said the responsibilities to name a successor "are bigger than any one party. They're about our democracy, and they're about the institution to which Justice Scalia dedicated his professional life."

UPDATE: 10:00 p.m. EST -- Obama ordered the U.S. flag be flown at half-staff in honor of Scalia. The order applies to the White House, all public buildings and grounds, military posts and naval vessels throughout the United States and U.S. territories.