The 18-year-old, who joined the Reds in the summer, went down in the closing stages of Tuesday night's third-round tie and had to be carried off on a stretcher.
Manager Arne Slot admitted afterwards that the problem looked severe, and on Wednesday it was confirmed that Leoni had sustained significant knee damage which is expected to keep him out for several months.
"It was not the debut I had always dreamed of, but I will give everything to be back playing in this magical stadium as soon as possible," Leoni wrote on Instagram, where he also thanked fans and team-mates for their support.
Liverpool players quickly rallied around the youngster. Captain Virgil van Dijk posted heart and fist emojis, while fellow summer arrival Florian Wirtz told him: "You'll come back stronger." Alexander Isak, who scored his first goal for the club in the same match, also sent a message of encouragement.
Jayden Danns, Conor Bradley, Ibrahim Konate, Andy Robertson and Stefan Bajcetic were among those who showed their support online.
The club's medical team are preparing for a lengthy spell on the sidelines, with Slot expected to provide a further update in Friday's press conference.
For now, Liverpool are left with just three senior centre-backs available for their Premier League trip to Crystal Palace on Saturday.
The injury comes at a testing time for Slot, who is already managing a congested early-season schedule.
Liverpool have taken 15 points from their opening five league games and will need to rely on their depth as they prepare for both domestic and European commitments.