Originally announced as a contusion during the game, Rangers manager Bruce Bochy confirmed in his postgame media availability that X-rays revealed the fracture for the All-Star third baseman.
"That's all I can give you right now," Bochy said. "He'll see a specialist, we'll see where we're at, what needs to be done, whatever. We just don't know yet."
Jung left shortly after turning a 5-4 double play. With runners on first and second, Miami's Jorge Soler ripped a sharp liner at Jung, who couldn't hang on to the ball. Jung then picked it up and stepped on third before firing to second to complete the double play.
Jung, who entered Sunday's game batting .275 with 22 home runs and 67 RBIs, is in his second season with the Rangers, and he has been a mainstay in one of baseball's best lineups.
Jung was 0-for-2 with a strikeout before leaving, and he was replaced by shortstop Ezequiel Duran in the lineup, while Josh H. Smith moved from short to third.
Duran homered in the victory, as did Nathaniel Lowe, Marcus Semien and Adolis Garcia. Andrew Heaney allowed four singles in 5⅔ shutout innings as the Rangers completed a sweep of their six-game homestand, a stretch that included the debut of starter Max Scherzer following his trade from the New York Mets.
"If you look at all facets of the game, we pitched well, and the offense, well, we had big hits and the long ball, certainly," Bochy said of the homestand. "We flashed the leather defensively, we played well there. When you have a homestand like that, you probably did play well in all facets."
Sandy Alcantara (4-10), the National League's reigning Cy Young winner, gave up five runs in six innings and allowed three homers for the first time since September. Alcantara is 1-4 in his past seven starts for the Marlins.
Information from Reuters and The Associated Press was used in this report.