Yes, you sure can. You can also have whiplash that cause pain to get worse as time passes. I would say get an xray and or mri.
Another consideration is whether somebody else is going to be considered at fault legally for your injury. You would want to be seen early if that is the case, and have an attorney involved, but let me tell you it can be hard to find an orthopedic surgeon to see you if he knows he has to wait for a lawyer to release money, and saying, "I will pay for it myself" didn't help at all where I worked, once you said somebody was at fault. Just so you know.
Most back pain after trauma goes away within 6 weeks. If you can take ibuprofen without problem, or Aleve, you can try to tough it out and see if it goes away on its own. But if you have shooting electric pain or numbness or you start dropping things or have foot drop or loss of bladder or bowel control, or something like that, then you of course need to see an orthopedic surgeon. You could make the appointment now, because there may be a 2-3 week wait, and then cancel if you think you are improving and don't need it.
I worked at an orthopedic office for a while. I was thrown off a horse once, and I just waited out the pain and I was okay. I was sure mine was just swelling and muscle bruising. The pain didn't make me think my spine was injured.