If I was visiting my sister and slipped on ice in the apartment complex's parking lot, can I sue the owner?

You may have a claim against the owner or person in control of the property.  I practice in Texas and to be successful on a claim like this you would need to prove that the owner or person responsible for the property knew or should have known that their was a dangerous condition on the property and failed to correct it or warn of the danger.  You would also need to prove that the dangerous condition caused you damages, for example an injury which required medical treatment, pain and suffering, etc.