Was I wrong to defend myself?

A friend's mom accused me of trying to steal the radio out of her 2015 Ford. The reason she thought it was me is because her dash was broken, and I had just installed a new radio in my pickup. I was furious. I told her this: "1. I'm not a thief. 2. I don't want your factory radio that is useless in any other vehicle. 3. If I was a thief, and for whatever reason I want your POS factory radio, I wouldn't 'try' to take it, I 'would' take it, and nothing would get broken in the process cuz I'm competent enough to take apart a dash." All my friends said I was overreacting, and the mom said I'm no longer welcome in her home. For me it's the principal, don't call me a thief, especially with no evidence. Don't sour my name. Am I in the wrong?

No way. It really doesn't matter how old they are, they shouldn't just accuse you. They should've calmly asked you, and it would be fine. If someone accuses you of something you didn't do, you can do anything you need to so you can show you're innocent.

You probably went about it wrong, what with all the insults, but no you were not wrong to defend yourself. Should have just stuck with "no, I did not try and take it".

You may have been right to defend yourself but you were wrong in your ill-informed attempt at character assassination. If you did not steal the woman's radio, you didn't steal it, and you were right to say so. But that's it; to insult her taste in electronics ("I don't want your factory radio…") or to impress her with your skill in thievery ("If I was a thief, and for whatever reason I want your POS factory radio, I wouldn't 'try' to take it, I 'would' take it…") is not only an unnecessary reaction but highly immature; that is how a child reacts, which leads me to believe that you are most likely a teenager, not an adult (you didn't give your age). What also bothers me about your post is that it is completely one-sided; you go to a lot of trouble painting her as The Evil One without bothering to explain why she would single you out in the first place. After all, aren't you her son's friend? If anything, you should have been at the the bottom of her suspect list, not the top. There's a reason for it, but of course you're not going to say.
Showing contempt and disrespect toward elder people, particularly when they are the parents of your friends, is never the right thing to do. If you are innocent, the truth will out without your having to step into the gutter in order to heave invective at someone. It has the opposite effect; it not only shows weakness of character but makes you appear even more guilty, which is precisely why she forbids you from entering her house.
She was wrong to accuse you of theft without proof or a show of evidence; you were wrong in defending your honor by trying to trash hers. This is a battle that nobody won; it's a stalemate.

Your mom looked in your direction for a reason. That is how it works. Your history would have been part of her thinking.
How you defended yourself means absolutely nothing.

Not at all.

Yes