How can I modify this voltmeter circuit to show 0.00V when nothing connected to the output
I think your values for R4 and R5 are backwards. That's the only way the numbers would make sense.

The only way to get it to read 0.0 Volts is to replace the op-amp with something that has a lot lower input offset. Take a look at the datasheet for LMV358 here: http://www.ti.com/product/LMV358

You can see that it has an input offset spec of 7 milliVolts. Your circuit has a gain of 14.66 (which is 100k / 7.32k + 1). So this input offset of the opamp of 7 mV * 14.66 = 0.103 Volts, so your 0.06 V output error is in line with what you should expect for this opamp. 

But before you even go replacing the opamp, the bigger question is: why are you attenuating the input signal by 14.66 and then amplifying it by 14.66? Your overall gain = 1. Why don't you just configure your opamp as a voltage follower?

That alone will reduce your offset error from 100 mV to 7 mV. Reconfigure your circuit first, and then look for another opamp if you still need lower than 7 mV offset error.

(P.S. Why do you even need R3 50 mOhm there?)