4 channel inductance meter with LM339
The topology isn't bad, but there are several critical problems:


Comparator input biasing:  Both inputs are held at DC ground.  You need to provide a small positive bias to the (+) inputs so the static state of the comparator  open-collector outputs is an open circuit.  This way the inactive comparators won't interfere with the active one.
The ringing LC tank circuit will go positive and negative, but your comparator inputs aren't designed for below-ground inputs.  The biasing I mentioned above can help here.
Resonant frequency: What value is this spring inductance?  I assume it's in the microhenry region.  You have 2uF capacitors.  This tank circuit will not ring enough to be useful, given the internal resistances of the components.  Look up "Q".
The 1N4001 diodes have a large reverse recovery characteristic, and will further dampen any ringing in the circuit.
In general, with (I suspect) microhenry-range inductors, you will only get useful ringing in the MHz region, using much smaller capacitors.  Neither the comparator, or the Arduino timer input are likely to work very well at these frequencies.