Ivancho, I wouldn't attempt to drive the MCT6 with a 1.5v battery. The LED would probably be destroyed. Driving it with a 1.5v logic signal would probably be OK, because the logic signal will probably have enough internal resistance (although the logic level may be degraded unacceptably). You seem to have interpreted the datasheet as a guarantee that, if you put 1.5v across the LED, the current will be 20ma. This is far from reality. The datasheet only says that if you force 20ma through the LED, the voltage will not exceed 1.5v.
Take a look at the
Fairchild MCT6 datasheet, which has more information. I have pasted the
typical voltage vs current curve below. Note that, at 25C and 1.5v, the current typically is over 100ma! This exceeds the maximum allowable current for the LED.
Keep in mind that the LED is a light emitting
diode, and as such, a small change in forward voltage will result in a large change in forward current. Another way of looking at it is, LEDs should be thought of as current-controlled devices. They should never be driven from a voltage source.
The reason I said that this part might work can be seen from the curve below, keeping in mind that this is a curve of a typical device. At 20ma and 25C, the voltage is only about 1.2v, so there is voltage headroom for a current limiting resistor. However, you might buy a device which has 1.5v drop at 20ma, so the same resistor would give you much less current, and therefore much less light output.