Is it possible to design a linear voltage regulator which delivers constant current but adjustable output voltage?
Simple: You don't.
What you're asking is physically impossible. In order to drive 1 mA of current through a 6 ohm resistance, you must have a voltage of 6 mV.
A linear current regulator achieves constant current by adjusting the output voltage in proportion to the load. So, for instance, if you replaced the load with a 10 ohm one, the regulator would increase the voltage to 10 mV to maintain the specified 1 mA of current. So, if all you're after is a constant current, that's the way to go. But just be aware that the output voltage isn't going to &quot;follow&quot; the input voltage, it's going to be reduced to whatever will provide the specified current to the load (subject to the device's operating limits, of course).