I am probably the perfect person to ask about this, as I've been through exactly this. My message will be long, but bare with me.
First, don't panic. When I got dismissed I hadn't yet gotten diagnosed/placed on meds. I knew something was wrong, as my academic performance steadily decreased, but I didn't know what. I always felt strange, but I could never put my finger on it. Turns out that I was undiagnosed with ADHD and MDD (depression) for years. So, while my case is a little bit different from yours, it's not much different. Just relax, realize that it's going to be a bit of a journey, but everything will be fine. I'm not sure what college/university you go to, so I'm not sure what their policies are, but I'm sure it can't be too much different.
So it's good that you're seeing a psychiatrist. If he's only monitoring your diagnosis and medications, start seeing a psychologist. You need therapy in addition to medication. It really helps out your case, and honestly you'll become a better person regardless of whether or not you think you need it (you learn a lot about life and what you want from it). The sooner you can get into therapy, the better.
Second, don't jump the gun. You can't get diagnosed and then expect to immediately get back into school. They want you to take some time to get yourself together. Most importantly, TAKE SUMMER CLASSES. Depending on whether or not you're on the quarter/semester system, the date for the first session may be over. Take the second session. Take one class, two if you can handle it. Study your ass off like your life depends on it. This is very important to getting readmitted, and they're likely not going to allow you back in this fall without it. If you don't, they'll most likely make you wait another year and take summer classes next year before they let you back in.
Third, submit your application right before the date. Give yourself a week to get letters from your doctors and write a personal statement. Explain your case, tell them what was going on and how you've improved/not going to make the same problems again. Submitting it close to the due date gives you more time for improvement, which they will want to see. Sometimes they don't allow you to submit extra information (notes, personal statement, etc), but in my case they allowed it but I had to take it to my dean's/advising office separately. Try to submit it together and if they don't let you then take your extra information to your advising office. Tell them that you submitted a readmission application and that you had some extra documentation that you would like to be considered. They most likely will allow it and give it to whoever is assigned your case.
Fourth, submit a retroactive withdrawal petition like "samuswashere" said. Most universities have this, so I'm assuming yours does. They're kind of difficult to get approved, and I'm currently in the process of one, but it doesn't hurt to try. Submit it with adequate information (letters from doctor describing severity, timing, duration, and how it affected your academic performance, and another personal statement).
Fifth, speak with an advisor. I know it sucks and it can be awkward, but these people are here to help you. The entire process will be SOOOOO much easier if you reach out to them. They can direct you in the right direction, and honestly you want them in your corner since most likely one of the advisors will be assigned your application.
I know the entire experience can be embarrassing, frustration, and anxiety-filled, but everything will be ok. As one of my advisors explained it to me, they don't dismiss you to punish you, but to keep you from damaging yourself. At the point of dismissal, it's obvious that something needs attending to. Whether it be medical/mental, personal struggles, or you just need some time to mature, they do it for your benefit. If not, then you would just continue to spiral out of control and trash your GPA even further. Readmissions aren't competitive, and they will most likely readmit you at some point if it's your first time. They want you to graduate, they want your money, and they want you to be successful. As long as you can fix your issues, prove that they're fixed, and show that you can perform acceptably academically, they're very willing to let you back.
Telling your parents is extremely difficult, and I don't know your parents so it may be different for you. My mom always suspected that something was wrong with me, but my dad assured her that everything was fine. Telling them was extremely difficult. There were a lot of tears and it was awkward and a bit depressing at the beginning. However, progressively they saw how much better I was doing and how much happier I was. They just want you to be successful, and while it's awful that it happened, they'll still love and care for you. They may have some hoops for you to jump through (I don't know if they're paying for your education or not), but just do it. Talk to them and work it out with them. It'll all be ok. I would wait until you see a psychologist and sign up for summer classes, though. That combined with sharing your plan for getting readmitted should make it a lot easier to handle.
Unfortunately there are a lot of people who don't believe in ADHD. It's a real affliction, as I'm sure everyone here can attest to. Medication makes a huge difference. I know my life has done a complete 180 since getting diagnosed and medicated. I went from being depressed and suicidal with no goals, motivation, or direction in life to having all of those things and much more. I know what I want to do with my life, I'm so much more motivated, I don't procrastinate and get things done. Life is so much better. It may be difficult to convince him, but he'll learn with time. Just keep in mind that this period of hardship will be over soon. Six months will fly by and you'll be in a much better place looking back and thinking about how much better things are.
If you have any other questions or just need someone to talk to about everything, feel free to message me. I've been there. I've been kicked out, diagnosed, medicated, gone through therapy, readmitted, and am no in the process of a retroactive withdrawal. Life throws you a lot of obstacles, but it's a long journey. In the grand scheme of things this is just a blip in your timeline. It's so much better to get diagnosed than later. I can't tell you how many people I've talked to who weren't diagnosed until their adult years and wished they had known in college.
Good luck, and just keep moving along. Take it one day at a time. Just try and make yourself a better you each day. Think about who you want to be, what life you want, and identify the steps to achieving it. Want to be ripped and get girls? Hit the gym. Want to be successful academically? Develop self-discipline and start studying. Read your books before class. Want to be successful at work? Look at other successful people and identify how they became successful. You can achieve anything you want as long as you're willing to put in the work. You can have the dream life, you just gotta work for it. Stay strong, it gets easier.