There's not a clear answer to that question. It mostly depends on how permanent your move is (4 years, alone, isn't enough) and what permanent ties you have to Ohio. Essentially, it's your call. If you use the foreign earned income exclusion, it probably doesn't matter much, as Ohio goes by the Federal AGI.
Turbotax (TT) is not doing because it's required only because it keeps transferring from previous years. Even though the OH return is prepared, just don't file it. You'll have it done if you end up needing it.
To delete it, see https://ttlc.intuit.com/questions/1900608-how-do-i-delete-my-state-return-in-turbotax-online