Scientists studying the field of tissue engineering are very interested in knowing the answer to your question. If we can understand the normal pathways of tissue healing, then maybe we can find a way for ligaments like the ACL to repair itself.
The poor healing capacity of the ACL can be explained in part by its biology. First there is a very thin lining or sheath around the ACL. Once this sheath is disrupted, the blood supply to the ligament is decreased greatly.
Normal healing and repair depend on the formation of a hematoma. A hematoma is a collection of blood cells trapped in the tissues after trauma or injury. Somehow the presence of the hematoma sets up the right environment needed for tissue healing. Without a blood supply, there can be no hematoma formation.
The hematoma provides a base camp so-to-speak for local growth factors and chemicals to come and set up a mesh or scaffold. Cells fill in around the scaffold forming collagen and scar tissue. It looks like there's a complex interchange between repair cells, growth chemicals, and the scaffold needed for healing. Without the hematoma to get the process started, ligaments don't recover on their own.