How long can I wait with back pain to see if it goes away without seeing a doctor?

No exact amount. Low back pain that doen't get better with over the counter meds in a couple weeks should probably get checked out. Sooner if there are any other symptoms like leg weakness, bowel/bladder changes, or pain that wakes you up at night.
See pain specialist. Each person needs to be evaluated on an individual basis. There can be several reasons for your pain. While it can certainly resolve on its own in several weeks, why wait. I advocate treating to decrease inflammation, reduce pain, increase function and quality of life. A thorough physical exam and diagnostic studies will help determine what treatment will be best depending on your diagnosis.
Back pain. Most back pain will go away on its own, but it can take anywhere from 1 to 6 weeks. If you can still get around, you'll probably be ok with just otc ibuprofen. Usually the only treatment a doctor provides is stronger pain medication. There is generally no reason to get an xray or other fancier imaging test unless the pain has lasted for 6 weeks or longer. If you can't move at all, see an md.