Thank you for trusting your question to JA today. I am a licensed attorney with over a decade of law practice and over 20 years of experience in the legal field. I’m happy to be of assistance.
There is an appeal process, but it's not like a legal "due process" appeal. This is because there is no legal right to military service, so it's not a court matter. The "appeal" is something called a waiver request. That is an appeal to the recruiting station commander to allow entrance over a disqualifying issue. While I understand that you say this is a minor issue in the past, it still can be disqualifying to the point that it requires a waiver to gain entry. It does not take much to be DQ'd.
As I noted, the waiver request isn't actually a "due process" right. It is a tool for recruiters to use in order to fill positions that they need to have filled. If this recruiter wouldn't process a waiver for you, that means they aren't really in need of anything. This is becoming universally true, as we are seeing fewer and fewer waivers processed or granted. That is a direct result of the still poor job market and the shrinking military budget. As a result, we have more people applying for fewer positions. Waivers become unnecessary in such situations.
So, to obtain a waiver, you have to find a recruiter with a need. If they have a need, they'll process a waiver request. This is no guarantee that it will granted, but the only way to even get it processed is to find a recruiter with a need. If the issue is minor, then there is certainly the chance that it would be granted.
If you have any further questions, please let me know. I invite follow up questions, so use REPLY for those. If you have no further questions then good luck going forward and please do not forget to rate my service with a top-three rating so that I receive credit for working with you today. Also, feel free to request me in the future, if you have questions concerning a different matter.