No they can vote either way. Each person running (GOP, DEMS, Green, etc.) has a set of voters that will vote. So whoever wins the state, their set of voters will cast the Electoral college vote for that state. A person in the electoral college can vote any way they so choose, however, it is pretty much political suicide for that voter to do so if they have any politcal office ambitions of their own.
It's also good to know that a Republican has never taken office without taking Ohio, which is a close race right now.
Based on his statement it sounds like a Republican could still lose the election even though they win Ohio. So not exclusively.
