Core strength (not just your abs) is what protects your spine (particularly the stabilizers, like the multifidi, erector spinae, transverse abs and obliques). These muscles do not work independently of each other. They all work together to keep your core strong and to support your spine properly.
Since you use your core to keep your body straight while doing push ups, it sounds like you need to build up your core strength.
More info would be nice. Age, weight, height, goals, how long have you been working out, etc.
And do you have a back injury now or have you previously injured your back?