Aloha & HOwZit!
Please keep-in-mind that what I am shraring is not-absolute but hopefully it will generate and motivate your interest to further seek healthy-information in reference to your "subjective-complaint" - back pain from doing push-ups.
Physical-POSTURE directes biological-energy flow or transportation.
When this "energy" is obstructed/restricted there is a "traffic-jam".
When we are caught in this "jam" we get stress-out and uptight - no different within the flow-of-life of your body.
Tension-Strain are two factors contribute to physical discomfort, soreness and PAIN when the correct exercise posture is not properly-set and utilized. Along with improper exercise-posture, one's breathing patter needs to be correct and compliments the exercise-actions.
From the prone-position or face-down, your feet-legs-hip must be in order which means spacing/distance, joint alingment as well as your should-arms-hands.
What affects one part of the body affects the whole body.
You might want to research proper-physical-positioning or have someone who has the right-understanding of the "push-up" exercise assess your overall exercise-performance.
With me, working with people on their "push-ups" I have witness that the lower-back-spine is not stable and the hips are not properly used to assist in stabalization and energy production.
Legs are to close together that cause a "closed-door" affect on the hip-thigh joints moving properly.
Although the "push-up" is seen as an upper-body strength-exercise, it is a body-weight resistance exercise that involves the whole-body and not its parts.
Going back to your problem, this is only suggestive advise and not an absolute cure-all.
Now that your sub-complaint is directed at your "back" (I am asuming it is more your lower-back area) the first action is to perform a stretching-exercise (that is to lay on your back) directed at the lower-back and after if you can get-up, apply "ice" to reduce inflamation the that area while taking slow-deep breaths.
Once this ice starts to give a numbing-tingling sensation, remove the ice and stay on you back and slowly perform low-back stretches and please do hold your breath. If your sub-complaint is upper or mid-back perform the same application.
If there is still persistence of you complaint after two-three day see your physcian/chiropractor to help with your complaint and let he or she know you were doing pushups and whatever exercise.
Never, again never second-guess you physical-health. You are not a "second-hand-human-being" take care of who is first in you life.
I aplogize if I went over-your head with my reply.
Go and start getting to know what's healthy when exercising and What Is Not Healthy in exercise.
Good Health To YOu!
mikey