Is the property owner responsible for tenant's damages who fell on the property grounds

Yes, perhaps. This is the very essense of insurance. More facts are necessary to answer this question. In my experinece, however, the insurance industry has done an excellent job in painting an unrealistic picture that the property owner will somehow will be fully responsible for damages. However, more often than not, business owners do have insurance and the protection against property injury claims are the reason responsible business owners (like myself) purchase business insurance. Recenltly, I have seen scare tactics on television commercials by insurance companies in which the insurance companies infer that the kids college funds wlll be taken away. This is borderline jury tampering as insurance companies large corporations attempting to make more money (sometimes at the ethical expense of not living up to the social contract of doing the what is right) by not fulfillling legal and contractual insurance obligations and underpaying the insurance contract (somtimes putting their insureds' at risk in some circumstances) when someone is injured and is found to be responsible for the claim. Unfortunately, in Ohio, a person is not allowed to sue the insurance company, nor is allowed to mention insurance at trial. Instead, they can only sue the corporation or person that has the insurance.