Ok, well to start, yes, mistakes happen to everyone. Including experienced nurses. Feel free to browse the site, search 'med errors' and you will see you are far from being alone in this.
The fact that nurses now work too many hours, don't sleep enough, have too many patients at one time, can't take breaks, etc, these are all reasons why we make mistakes. Unfortunately, without support from management, these things can't go away to reduce risks.
If this was your one and only mistake, why would you think it makes you a bad nurse? Does being a nurse mean you lose all rights to be human and make mistakes? I think that after all this time, the fact that you are still obsessing over it and questioning yourself means that you genuinely care about your patients and their safety and well being.
I also got fired from my first nursing job straight out of school. It was in the best mother-baby department in the city, and I was ecstatic to have gotten the job. I had excellent recommendations from all my nursing teachers. When I didn't manage to fit in with the cliques, the other nurses started nit-picking everything, right down to going behind me to ask my patients how I had done. Although I had had MANY patients tell me they loved me, once they started looking for problems they seized on one negative review (I was busy with a baby not doing well, and 'neglected' a new mommy who thought i should be in her room more often). Since I was still in my probation stage, they fired me.
I was devastated and like you, questioned my career choice. I worked as a waitress for a while to make ends meet. To top it off, my former nurse manager was giving me bad references for jobs I was applying for.
I kept at it, and ended up in a job that I loved. I had a difficult time initially getting over the previous criticisms, and it showed in my performance at first. But with help from colleagues, it got better.
Now, will it be hard to put it behind you? Most likely. But hey, use it as a learning experience. When you do end up working in nursing again, you will pay that much more attention to what you do. You will make darn sure not to let yourself get distracted when you need to focus on a task.
You'll be fine. Just don't give up!