120v Inverter and GFCI
I get 120v from neutral to hot to neutral and 60v from neutral or hot
  to ground.


The important thing to note here is that you \$\color{red}{\text{get 60v from neutral or hot to ground}}\$

This tells me that the output is likely floating BUT loosely coupled to ground with EMI capacitors of a few 100 nF. Something like this: -



This means that a regular GFCI won't work because you are just not able to drag the current through the resistor. It's a bit like connecting a battery to ground via a 1 kohm resistor - no current will flow because you haven't made a DC circuit.

Your inverter's output is galvanically isolated from ground hence a single point connection from either hot or neutral to ground won't cause a significant current draw that would trip the GFCI.