IAA Fan, I don't want to argue over something that I care little about, but the state of Ohio gives no money to private colleges. Students at those places can receive government financial aid, which is why those institutions must abide by certain standards. But a direct subsidy like a state school....no.
On the K-12 level, the state government gives nothing to Catholic schools, except for bus transportation. You are thinking about the voucher programs that enable kids in dysfunctional public schools to attend a private alternative. Many use those vouchers for Catholic schools. That is the source of confusion about this topic. There is no direct state aid to Catholic schools.