I'll take a stab at this off the top of my head. NO.
rpm equates to angular velocity, not the 'straight line' velocity of an object in orbit. The centripital force actually creates an acceleration of the body, i.e. it changes the direction of the body. For instance, if the ball being swung in a circle on a string were to have it's string cut, the ball would tend to travel in a straight line (given no earth gravity effects). Thus, the string is changing the direction (accelerating the body) constantly. The tension in the string is equivalent to the force changing the direction, which is the centripital force. Given a higher RPM, there's also a higher 'straight line' velocity that has to be overcome by the tension in the string.
In other words, f=ma, and a higher rpm requires a greater force to maintain the equivalent centripital acceleration.