Rotate a brush less motor using arduino only
By brushless motor, do you mean a BLDC (brushless DC) motor? If so, no this is unfortunately not possible. You need to synchronize your phases to actually rotate the motor. Hence the ESC or an h-bridge with your own code. On top of that your Arduino doesn't have the power, nor the safety circuits in place to prevent the energy in the motor from not killing the board when you turn it off. 

The servo library is used for, lack of a better term, the hobby servo motors. 

http://www.instructables.com/id/BLDC-Motor-Control-with-Arduino-salvaged-HD-motor/

Here's an example. You're looking to do the same. Good luck!