Pushups work much more than just your chest, particularly your core muscles, critical to everything including your back. Try just a static plank for ten minutes, and tell me what you feel.
For my low back pain, I stretch the hip flexors to relieve their pressure on the lumbar spine. I strengthen my abs with side planks to stabilize my core without engaging the hip flexors. Finally, I strengthen my glutes.
Every situation is different, and not all back pain should be treated with a pushup, so you have to understand body mechanics so you're working on stretching or strengthening the right parts for you. Going to a Physical Therapist is very helpful, then progressing to some appropriate strength training.