How independent is the output voltage from the load resistor in this linear regulator?
There are two types of regulation specified in data sheets for regulators ...

Line regulation - change of output when the input changes
Load regulation - change of output when the load changes

You are talking about load regulation.

Your equations for the output voltage assume infinite gain in the opamp. Once you've written down the full equations including a finite gain, you'll find that the voltage at the junction of R1 and R2 will have to vary to provide the varying current output, and there's your change of output voltage.