“Can a cat's sprained leg heal on its own?”

A cat's sprained leg can heal on its own given at least 2 weeks of bed rest. If the pain is extensive, I would recommend contacting your primary care doctor to prescribe pain medication. I would also recommend contacting them to perform additional diagnostics to ensure that the injury is not just a sprain. Missing fractures, torn menisci or ligaments would prolong and cause improper healing.