I do not practice in Ohio, but I believe this advice would apply here.
The question isn't "Can I sue?" Generally in the United States you can always file suit. However, you might not be successful in obtaining compensation. That is the real question, "Does my case have enough merit that I'll be compensated for my injuries?"
Property owners are responsible for injuries caused by "dangerous conditions" on their property, which they either knew of or would have know of after a reasonable inspection of the property. In this case, the ice may be a dangerous condition on the property. However, the weakness of your case--as well as many premises liability cases--is whether the property owner had sufficient knowledge of the dangerous condition to either fix it or warn others prior to your fall.
The facts--as always--provide support for both you and the property owner. The parking lot had been plowed the day before your accident. This supports your argument that the property owner had knowledge of the dangerous condition prior to your fall. But it also supports his position that he made an effort to take care of the icy pavement.
I believe you'll be able to find a lawyer in your area who will take your case.
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