Consider this, the IRU along with the ADC computes the following:
Position (lat long)
True & mag heading
Winds and direction
Attitude
Velocity
Accelerations
Angular Rate Data
Altitude
The IRU only uses ADC data for winds, altitude, and velocity.
The main purpose of the IRU is to sense and compute linear accelerations and angular turning rates about the ariplane's roll, pitch and yaw axis.
*Data taken from B737 Classic AMM
So yes, the IRS is standalone. But for alignment, valid ADC data is needed...or else alignment will not complete. Once aligned, ADC data is not needed for the IRS to do it's job.