Could I run a motor that is rated for a 6-7" propeller on 3-5s battery system with a 9 inch propeller. Would it be possible to do so without damaging the motor, or would the torque required be too much for it to handle even with power draw being under the limit.

The motor I have in mind is the Emax Eco 2807 1500 Kv. It's has a max rated power of 1200 watts and recommends a 6-7" propeller.


A 9 inch prop I tested with a DYS 3536 1450 Kv motor only drew around 300 watts. I assume power draw should be similar, would it work?

Sure. You could run a 12" prop.
(If the pitch was zero, and the prop was light)

I'll wait for other responses, and proceed only if desired...

I am really interested in this topic.

As much as the size of the prop matters, the pitch of the props matters too.
The size from 6" to 9" is substantial difference in the load, but different sizes will have different load like 6x4 and 9x3. Even 2 blade and 3 blade prop will have different effect.
You can run the motor if the amp draw is less, but on the long run, it will wear and tear the motor.

It does not have a simple answer....best would be to try out different combinations of prop dia, pitch and bty cell count... there are too many variables

1. Size/Dia.. a increase from 7 to 9 is nearly double the disk size (49 vs 81) .. so there will be a substantial increase in the torque required to spin it to the same rpm.
2. If the prop stalls or cavitates , thrust will be very low... this can happen if the prop rpm is too low or too high. And it is also dependent on the pitch of the prop. Furthermore the pitch speed of the prop (which is directly proportional to the pitch of the prop) is generally designed keeping in mind the stall speed of the aircraft.
3. the motor could also stall (stopped / rotating slower, like when a external braking force is applied) in which case it will draw more (huge) current and burn.... Though the wattmeter will show higher wattage, thrust will be low and it might burn the motor or esc......Here a lot depends on the type and quality of the winding and the materials used
4. Bty cell count has a direct impact on the rpm .. motors are designed to work best within  a defined cell count range 

In some of my planes I have reappropriated and used high Kv quad motors ...  .. like in one, for 2200kv motor designed to turn a 5x5 prop on a 4S,  I use a 2S bty with a 7x3 prop......It is quite a trial and error, .... sites like "eCalc" have historical data of some motors captured and can simulate outputs based on different prop dia, pitch, cell count. You can try and see if the motor you are using is listed there