Whether an employee, either former or current, is entitled by law to see their personnel file is state specific.
You may always ASK. But in Ohio, even a current employee has no right under the law to see or have a copy of their personnel file. There is no harm in asking, but if the employer refuses there is nothing in either Federal or Ohio law that is going to force the employer to release it, short of a subpoena.
No conclusions should be drawn by the employer's refusal. Many people are tempted to conclude that, "since they won't let me see my file, that must mean there's something in it that will prove I was illegally treated". For this reason I always advise employers to allow employees access to their file when they ask. However, many employers believe that the files are company property and that no employee should be granted access, regardless of how squeaky clean the file may be.