Why doesn't a single engine airplane rotate along the longitudinal axis?
First, it's about torque, not power.  A helicopter rotor has a large amount of torque at a relatively low RPM.  A single engine airplane has much less torque at a higher RPM (Power = Torque X RPM).  On top of that, for a craft of similar size, the helicopter will have much more power (it has to lift the copter directly.)  So the helicopter has to deal with torque levels several multiples higher than that of a single engine aircraft.  

As to the solution, most single engine aircraft have the vertical stab offset by a small amount to generate a counter force to the torque.  If you look at some high power aircraft such as the P-51, it's a noticeable offset. And that's set for cruise power.  Takeoff power requires additional rudder input.  

It should also be noted that for smaller aircraft, P-factor is a bigger turning force during climb than engine torque.