A constant current driver will raise the voltage until it is able to force the mentioned current through the load. Voltage can't rise to infinity because of circuit limitations/configurations.
Your driver will want to push 600 mA irrespective of load connected to it. In your case, a 120 ohm resistor, it needs to set the voltage to 120x0.6 V = 72 V.
72 V is clearly out of possible voltage range. Now this out of spec operation is handled differently by differently designed drivers.
Case 1 - Driver will max out at 38 V and push 38/120 A = 316 mA of current continuously.
Case 2 - Driver will try it's best to reach 72 V but will fail and reset. This will keep on repeating. If you put an oscilloscope, you will see a PWM sort of curve. ON time represents the driver trying to reach 72 V mark, OFF time represents the reset duration where driver is waiting for the anomaly to go away before it tries again.