Watson suffered the injury as he jumped with the Reds striker during the first half of Saturday's Premier League contest, which Liverpool went on to win 3-0.
In a tweet, Sterling wrote: "To Ben Watson I didn't realise it was serious as that. I'll be praying for a speedy recovery mate. I'll have you in my prayers every day."
Sterling, 17, was absolved of any blame for the freak injury by Wigan boss Roberto Martinez, who said: "Both were looking at the ball - it's a real accident. It's a nasty collision."
Watson, 27, is expected to be out of action for four months.