Low Voltage between Neutral and Ground wires in Dishwasher Junction Box

The green wire ends up connected to the chassis of the washer through the connector on the gray wire. If you'd open up the dishwasher you'd see one of the connectors going to a metal plate.

If you have a multimeter/continuity tester you can test this by checking the continuity between one of the connectors where the grey cable plugs in at the dishwasher and the chassis. The corresponding connector will be connected to the green wire.

This is done to reduce mistakes and questions from people who don't know here the green wire goes.

Besides that the black cable should be clamped down with a strain relief to remove stress on the copper wires. That would lead to bad connection arcing and general fire hazard.

The dishwasher does not know if you have a ground or not. The ground wiring on the inside of the machine (the dishwasher in this case) is tied to everything that will conduct electricity easily and could come into contact with the user; typically anything metal.

A ground wire can be thought of as simply a backup neutral. So if a hot wire ever touches some part of a metal system, the current will have this backup wire to run the current back to the breaker box. The breaker will detect this increase in current (unimpeded path) and trip which will cut off power to the device. If this wire weren't here, the metallic object would be energized, but it would not trip a breaker. It would simply be waiting for something (a person) to come along and touch it to give it a better path to ground.

So, as long as a hot never comes loose, you wouldn't need the ground. But if it ever were to, this wire could save a life (or prevent a really strong tingling sensation).

To shorten things off and end this; you don't need the ground for the dishwasher to function. It will work perfectly fine without one and you just wouldn't connect the ground wire on the other end. You then run the risk of a wire coming loose though.

But, what should be done regardless and will also provide grounding functionality is to use a GFCI breaker. The breaker would replace the existing breaker feeding the dishwasher and include a pigtail to the neutral. This breaker detects the current that is feeding into the circuit and out of the circuit. As long as everything is running like normal, it will work like a normal breaker. However, if a hot ever comes loose, the GFCI would detect that current is going in, but not as much is coming out and the breaker would trip itself.