The teenage midfielder sustained the injury after an initially innocuous-looking challenge by Pascal Struijk, who was shown a straight red card after VAR intervention.
Elliott was given lengthy on-field treatment before being carried from the field on a stretcher, while his injury had clearly upset both sets of players and benches.
The 18-year-old said on Instagram on Monday: “I’m of course absolutely devastated with what happened yesterday at Leeds, but have been totally overwhelmed by the love and support shown to me by the entire football world following the injury.
“Thank you so much to everyone who has reached out or sent messages to me and my family, it means so much to us.
“Also a massive thanks to everyone inside Elland Road for the reception you gave me immediately after it happened.”
Elliott was discharged from Leeds General Infirmary and while the full severity of the damage has yet to be confirmed, he will undergo surgery.
The former Fulham player added: “I am now fully focused on my recovery and will give everything in my rehabilitation to be back out there as soon as I can.
“I know I have an incredible support network behind me at Liverpool and together we will get through this.
“To all of the Liverpool fans, your support means the world to me. I’m one of you and I can’t wait to be back faster, fitter and stronger to help the team in the future.”
Elliott has emerged as a key performer for Jurgen Klopp’s side this season after spending last season on loan at Sky Bet Championship side Blackburn.
He stepped off the bench in Liverpool’s opening-day win at Norwich and after making his full Premier League debut in last month’s home win against Burnley, has started against Chelsea and Leeds.
At Elland Road on Sunday, Elliott twice threatened to add to the scoring in the first half before Struijk’s tackle in the 59th minute.