Assuming a resistor is in series with a diode (say, a 1N4001) so that the diode has around 1mA of current, will its forward voltage be usable as a voltage reference to within 1% (relative to itself over time and with ambient temperature changing by 10F max)? The voltage doesn't matter, as long as it doesn't change to a noticeable degree.
Is a diode's forward voltage stable enough to be used as a reference voltage?
No, it will change with temperature: -
Picture taken from here
At 25 degC and a bias of 5.1 mA (for example) the voltage will typically (that's typical and not always) be about 627 mV. At 30 degC the voltage will have dropped to about 618 mV - that's a change of 1.4% for a minor difference in temperature. In voltage reference terms that's a stability of 2878 ppm/degC. Compare this to just fairly cheap voltage references having a drift of 50 ppm/degC.
Assuming a constant, preferably low current (1mA) is applied to a diode
If you have a constant current source then apply it to a resistor and you'll get pretty much stable results compared to the diode.
I have seen some instrument schematics, that use LED diode as power indicator and voltage source. Could the LED be a better alternative? Search for current, voltage source schematics examples, perhaps you will the answer.
P.S: The "instrument" is the LeCroy scope differential probe, not a toy.
You can design BandGap voltage references, using multiple diodes running at different currents (thus different current densities and thus difference temperature coefficients. Some opamp circuit is needed to extract the CTAT. You combine the PTAT and a proper amount of the CTAT, and extract a very stable voltage reference. If you do this, please honor Paul Brokaw for the invention.