Driving a Turntable with a Two-Phase, Bipolar Stepper
You could do this with linear amplifiers such as high power op-amps, but that's not what would commonly be done today - it's very challenging (/expensive/inefficient) to build high power linear amplifiers.

Instead, it is more common to use full-on/full-off switching devices such as MOSFETs (or at very high voltages, IGBTs) in motor drivers.  These are switched at a rate many, many times the commutation frequency, with a varying duty cycle for each pulse.  The duty cycle varation - called pulse width modulation - effectively approximates the desired sine wave.  Since the actual switching frequency is faster than the system can mechanically respond, it ends up averaging it out*

That said, a stepper would be an odd choice for a spindle motor.  More common would be to use a brushless AC servo motor.  Much as with a stepper this would be driven by several distinct phases of PWM-approximated sine waves.  A main difference is that a servo motor typically has fewer poles - in effect fewer "steps" (electrical cycles) per mechanical revolution - and so needs a lower commutation frequency for a given rotational speed.  

*though some brushless motor controllers intentionally modulate the applied voltage at audio frequency and make the motor itself emit audible beeps while you work through their options menu.