This isn't a diagnosis or a strong opinion by any means, but you do come across quite 'off' with that post. Just maybe the psychiatrist evaluating you picked up on that? "Ruining your life", feeling as if you're becoming mentally ill, going to explode if you can't talk to anyone etc. You're right, this really isn't the right forum for solving your particular problem with a foreign army. Yet you posted it here anyways.
For the subject matter, try finding someone knowledgeable on Reddit, there are plenty of subfora there which might be relevant. More importantly, if this is actually "ruining your life", get a specialized lawyer. This is the obvious and most important advice. Noone here will be able to tell you with certainty about the appeals process (if there even is one), the filing deadlines and what have you. You need a legal expert. So go pay for an expert.
Hope things will work out for you.
So I've been rejected for conscription in the IDF because the psychiatrist thinks the Asperger's diagnosis I received as a child means that there is something wrong with me. Never mind that I've been examined very recently and been recommended for enlistment, he thinks that even though I probably don't have Asperger's, there must be something wrong with me because in the past I've had trouble socially. Of course I have no such problems now, but it's not as if he's going to risk his job in the face of anything less than perfection.
(This, btw, is what I meant when I said there was no such thing as a competent mental health professional- the entire system works against evidence-based methods.)
There has to be something wrong with this, some way that I can appeal. I have no idea of the Israeli legal process and I'm not sure if I could just write a letter to someone, or if I might need a lawyer. I can definitely prove that there is nothing psychologically wrong with me. I just have no idea where to turn, no idea how to do anything, and have no allies whatsoever. I feel like my life is collapsing, and I do have very good reasons personally for wanting to join the army. It's not just something I felt like doing.
This community obviously has better things to do than this sort of thing. But I feel like I'm going to explode if I can't talk to anyone, or get some idea of what I can do. I feel almost as if I'm becoming mentally ill.