Yes, your employer can fire you for absenteeism. You should have followed your employer's rules for requesting time off. I doubt that making your own unofficial arrangements for others to cover your shifts qualifies - employers typically want you to ask your supervisor for permission.
As others have already mentioned, if you were injured on the job you need to tell your supervisor that ASAP (you really should have done so when it happened) and apply for workers compensation. Here's a helpful webpage about workers compensation, courtesy of Illinois Legal Aid Online.
If your injury was not on the job, then you MIGHT be eligible for unpaid FMLA (Family and Medical Leave Act) time off. A lot of rules apply. Your employer must have at least 50 employees within 75 miles of your work location. You have to have worked there at least 12 months and have put in at least 1,250 hours during the past 12 months. And you have to have a health condition that (a) prevents you from performing the essential functions of your position AND (b) is "serious." You will need a doctor to fill out Form WH-380-E for you. If you did not follow your employer's rules for requesting time off, your employer might be able to delay or deny your FMLA leave.
If you think you might be eligible for FMLA leave but aren't sure your injury is considered "serious," come back on this forum and say so. I'll try to find more information about that, as well as whether it's likely your employer can deny your request because you didn't follow the rules for requesting the leave.
Of course, maybe you don't want to bother finding out more about FMLA leave, since it's unpaid leave and your employer could just wait a month after you come back to work and then terminate you for some other (fake) reason.
I dont have insurance so I haven't been able to go to the doctor.
Have you googled your injury to see what kind of treatment it might need? If by delaying a doctor visit you risk doing yourself permanent damage or greatly delaying the day you can start working again, I'd call that a bad decision. I have had decent luck actually shopping around by phone - I call and explain that I am a self-pay patient and ask how much I can expect to pay. If they are evasive or the number is high, I thank them and hang up and call another doctor (or urgent-care center). But if you have no money your best bet is probably trying to get on Medicaid.