I'm 17 and I work the whole day
I want to say that due to your age, they probably just found someone else and let you go. I think "fired" is a harsh word, and chances are your manager is probably immature and just wanted to be the one who had the last thing to say. I wouldn't really stress it all that much.
Also I want to say most retail shops and fast food restaurants I know of never requested a 2 week notice. When I was 16, I left my first job by just calling them and saying I didn't want to come to work no more. While not professional, they didn't really say anything other than "okay." Nearly everyone I knew from there, at the time, just up and left as they saw fit. The only time I ever gave notice was once at this retail department store I worked at. That was my last job before getting a "real job" and I didn't even get a good bye. Just got told they got someone else scheduled and I didn't need to come in no more.
A good lesson is to get everything written. If you only spoke verbally to someone, maybe they didn't pass it down. Have it written down and given to your manager that you intend to leave the workplace.
As I said though, I feel like at 17 you have nothing to worry about though. Just do things as you see fit for now, but learn as you go. Right now you learned that you should write down your 2 weeks notice and give it to your boss in person. That way everyone knows and no one is surprised.
I wanna know that what could I say to prove that he couldn’t fire me on my last day cause I already gave him my 2-week notice, that I was gonna quit and go to another job yet he still fired me, I'm concerned and I need advice.
The problem is in most states in the USA, employers are considered at-will meaning that outside of any protected reasons, they can fire anyone for any reason. They could say you didn't come in on your last day wearing purple shoes and fire you citing employment agreement.
With that said, I wouldn't really say anything at all in your new work place. You got the job. They're not going to double check a 17 year old kid's resume and bother to call up their last place of employment.
Also a huge thing is when you're under 18, generally most places don't bother to check because most background investigation don't look at records under 18. I also want to say that since a background investigation is a legal document consenting their investigation, you can't sign it without your parents approval. I don't think a single state allows a 17 year old kid to sign a legal contract agreement for consent.
Also of importance that most places hiring a 17 year old kid knows that there is a very high turn over. This isn't like the movies where you work at pizza hut for the next 3 years before going to college. No, they know that you're probably going to quit within the first couple of months.
So don't worry. Go to your new job and enjoy it.