Jackson left in the first half of Monday night's game against the Carolina Panthers.
Head coach Andy Reid made the announcement about Jackson's injury during his Tuesday afternoon news conference.
Reid said Jackson injured multiple ribs "up high" and was placed on injured reserve.
Later in the day, the Eagles announced Babin's release in an emailed statement.
A statement attributed to Reid read "We appreciate everything that Jason has given this team over the last couple of years. We wish him all the best as he continues his career. By releasing him today, this gives us an opportunity to give more playing time to some of younger guys in the defensive line rotation."
In other injury news, Reid said running back LeSean McCoy has not progressed in his recovery from a concussion, and quarterback Michael Vick - also suffering from a concussion - has not yet passed the IMPACT test.
T Jason Peters, who had not recovered from his Achilles tear, was also placed on injured reserve.
The Eagles lost to the Panthers 30-22, dropping to a 3-8 record after losing seven straight games.