Will a purely inductive load damage my function generator?
It should be just fine. 

Your function generator will look like a low source impedance 400mV p-p source (like an op-amp output but beefier) with a 50 ohm resistor in series, so the complete circuit is not purely inductive (not that it's a problem).

The inductor has an impedance of \$2 \pi f L\$ for a sine wave or about 7.5 ohms. 

Any decent commercial function generator should not be damaged by even shorting the output or connecting it to arbitrary impedances, and these levels are pretty benign. 4mA peak into even a short circuit is nothing to be concerned about. Don't try a 100uF capacitor charged to 300VDC though.  Your function generator can supply 100mA peak into a 50 ohm load.