Was hired into a company and a month later I cut my finger with a cutter at work. They sent me to get stitches then drug tested me and found cocaine.

Ohio is an at-will employement state, meaning that if you do not have a written contract of employment with your employer, then you can be fired for any reason or no reason at all (other than for protected discrimination- i.e., age, gender, race, religion, etc.).  So long as you did not have a written contract of employment, then legally the employer was allowed to terminate you for any reason or no reason.  If you did have a contract, then you need to review the contract, as the terms of the contract control regarding termination. Therefore, even if the drug test produced a false-positive, the employer legally can terminate you, as they do not need a reason to terminate you (unless you had a contract). Best of luck.