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.
While I agree your core is critical to your back, a pushup or plank also strengthens your hip flexor muscles. A couple of those (e.g. psoas) attach to your lumbar spine, so strengthening them makes them shorten and pull harder on the lumbar spine, increasing your pain. For low back pain, it is the wrong exercise.
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.
