First you need to get your basic electronics terms clarified a bit. There's no such thing as a motor "absorbing" the voltage of a battery. When you connect things in parallel, all branches have the exact same voltage at their ends. What there is more or less of is the current through each branch. Lower resistance, more current.
While it is true that the motor presents a much lower resistance to the battery than the branch with the LED, this only affects the *ratio* of the currents going through the motor vs the LED. The absolute value of the current through the LED should stay the same as without the motor attached, as long as the battery is still actually providing 9V as before.
I think what's really happening here is that the motor is too great a load for your 9V battery. If we assume about 3 ohms of resistance for the motor, the current it draws is around 9V / 3Ohm = 3 Amps. That's quite a respectable amount of current, and likely beyond what those typical consumer-grade blocky 9V batteries are intended to provide. Check the voltage at the ends of the battery while it is powering the motor. It will likely be much less than 9V, probably even down to 4-5V (especially if the battery is nearly depleted) which might not be enough to light the LED.
The 0.3 Volts figure seems highly suspect. Is this with the battery connected, or is it when you are spinning the motor manually? If it's the latter, buying a 9V or 12V motor will not help much, since they'll only give you at best twice the efficiency, whereas it is clear you need 20 times more. As mikerj said, try spinning the motor faster, much faster.