Gibson was carried off on a stretcher after injuring his knee in a first-half challenge during the Republic of Ireland's World Cup qualifier against Kazakhstan in Dublin on Tuesday.
The 25-year-old was assessed by club medical staff on his return to Everton and, following confirmation an operation will be required, a lengthy lay-off is likely.
Everton were coy about how long he will be out, but damaged cruciate ligaments usually require at least six months rehabilitation following surgery.
A statement on the club's website, evertonfc.com, read: "The length of time the 25-year-old will be (out for will be) best determined once the surgery is completed.
"However, the scans did confirm to Everton's medical team that no additional damage was sustained to the structure of the right knee, ensuring all treatment and rehabilitation will be focused on repairing the damaged ACL."
The injury is a fresh blow for the player after he missed the start of the season with a different knee problem.
His return to fitness, to feature in his first Barclays Premier League game of the season against Manchester City on October 5, preceded his recall to the international scene last week.
He played 90 minutes against Germany last Friday but lasted just 37 against Kazakhstan, a match which was watched by his club manager Roberto Martinez.
After the game King said the injury "wasn't nice" and now Everton, who have made a positive start to the season mostly in his absence, will have to continue without him.