If you're healthy and using a normal, dynamic rope, no it won't. It won't even hurt. A factor 2 fall onto webbing or dyneema can totally mess you up though, even from a couple of feet.

The UIAA (climbing standards body) disagrees with you. The specs of climbing ropes are set by them to result in no injury in worst case falls. And plenty of people have taken factor 2 falls with no ill effect.

And industrial fall arrest literature disagrees with the UIAA. A 4kN loading will break your spine in the correct orientation.

Workplace exposure limits are always lower than injury thresholds.

You'll want to retire that rope, but nope, I've had a couple of high fall factor falls with no ill effects. A true fall factor 2 fall is impossible to achieve in a real setting (knot, etc.), but I've taken a prob 1.6 or so.