The 87-year-old, who died of complications from metastatic pancreatic cancer, served on the nation's highest court for 27 years, becoming a liberal icon and a leader of the court's left-leaning wing.
"Rosalynn and I are saddened" by Ginsburg's passing, Carter, who at 95 is the oldest living Democrat to have occupied the White House, said in a statement. "We join countless Americans in mourning the loss of a truly great woman. We will keep her family in our thoughts and prayers during this difficult time."
As a jurist, Ginsburg's milestones on the court included a 2004 majority opinion blasting Texas prosecutors' behavior in a capital trial that the court determined was riddled with errors, and a dissent to a high-profile ruling throwing out Michigan's controversial affirmative action law in 2003.
Among the most important rulings may have been in U.S. vs. Virginia, the 1996 decision in which the court ordered that women be admitted to the previously all-male Virginia Military Institute.
"A prime part of the history of our Constitution, historian Richard Morris recounted, is the story of the extension of constitutional rights and protections to people once ignored or excluded," Ginsburg wrote in her majority opinion, finding no merit to arguments that admitting women would "destroy" the school.
"A powerful legal mind and a staunch advocate for gender equality, she has been a beacon of justice during her long and remarkable career," said Carter. "I was proud to have appointed her to the U.S. Court of Appeals."
Fox News reporter Bill Mears contributed to this article.