“Can torn ligaments heal naturally?”

Yes. A brace is the usual treatment. Sometimes there may be residual instability

Unfortunately no. Ligaments & tendons have poor blood supplies, which means they don't heal well. They scar down, which is weaker & less flexible than regular tendons. Look into Regenerative Medicine options to fix the tendons without surgery.

Complete tearing may need surgery, and partially torn managed by proper immobilization.

To an extent. It depends: if they are completely torn, it's not likely they will heal completely.

Once the body sustains an injury, it naturally begins to heal itself. However, there are three grades of ligament tears. Grade 1 is a mild year, which may heal with proper rest and immobilization according to guidance from a health care professional and gradual return to activity. Grade II is a moderate tear and most likely would need guided rehab and gradual
progression to activity. Lastly, the grade III is a complete tear of the ligament which would need intervention in facilitating healing. This could be a repair or grafting depending on the part of the body that the ligament tear occurred. Regardless, you should have the ligament tear assessed by a qualified professional.
I hope this helps.

All depends on the injury but they can heal