Rizzo also revealed that Eaton had suffered a torn meniscus and a high ankle sprain.
No surgery has been scheduled, Rizzo told reporters, adding that he expects Eaton to miss six to nine months.
Eaton, 28, suffered the injury in the ninth inning of Friday's 7-5 loss to the New York Mets on a bang-bang play at first base. He hit a roller to the left side of the infield and just barely beat the throw to first. After reaching the bag, he collapsed in a heap, and the crowd at Nationals Park fell silent.
"It's probably the worst pain I've ever felt in my life," Eaton said. "When I heard it pop, I thought it was my ankle, and it was my ACL; it wasn't my ankle. I looked down to basically see if I saw any bones and make sure that everything's still there."
Eaton wouldn't rule out a possible return in 2017.
"I'm going to work my butt off and give myself the best-case scenario," Eaton said. "And to play this year would be great, and if that is the case, that means we are playing [postseason baseball] in October."
The outfielder, acquired in a blockbuster December trade with the Chicago White Sox that included the Nationals' top two pitching prospects plus a 2016 first-round draft choice, was hitting .297 with two homers and 13 RBIs. His .393 on-base percentage is the second-best among leadoff hitters (minimum 50 plate appearances from that spot), trailing only Toronto's Kevin Pillar (.418).
Since his first full season in 2014, Eaton has been one of the most valuable outfielders in baseball, with only four others accounting for more Wins Above Replacement.
Trea Turner, last year's runner-up for National League Rookie of the Year, will return to the leadoff spot in place of Eaton.
"We feel we have the personnel specifically at that position to compensate for him," Rizzo said. "It's one of the greatest depth positions we have in the organization, so we feel good about that."
Michael A. Taylor will get the first shot at replacing Eaton in center field, though manager Dusty Baker made it sound as if Taylor will need to start hitting better to keep that job. Taylor entered Saturday batting .095 (2-for-21), but he raised that to .192 by going 3-for-5 while hitting second behind Turner.
The Nationals placed Eaton on the disabled list Saturday. To take Eaton's spot on the roster, Washington recalled outfielder Rafael Bautista from Triple-A Syracuse before Saturday's 5-3 loss to the Mets.
Information from The Associated Press was used in this report.