No, no, no!
It's an easy error to make, but you are thinking wrong-headedly about what that green wire is for. We call it "ground" but mainly it's to provide a return path for electrical faults. A path that isn't you. The return path only works if it goes back to the electrical service panel.
As a separate matter, we want this "equipment safety ground" and the neutral to be quite near whatever ambient voltage may exist in the soil around your house. That way equipment safety ground can't shock you if you're standing in a puddle. That is the only reason for the grounding rod.
Now here's where people get confused: thinking these are interchangeable. Actually, earth is a terrible conductor. You would not want to use earth as a ground wire. If you had a fault from hot to ground, the fault would try to "lift" the equipment safety grounding system up to 120V. The high resistance of the earth would mean it would succeed! Not enough currernt would flow to trip the breaker. Your entire grounding system would be energized near 120V which would make even the cover screws on a light switch deadly.
That is why you absolutely need the equipment safety ground to be an actual conductor, which goes back to the service panel. That way, there is a fat, low-resistance grounding path, which assures that a ground fault will flow a lot of current and definitely trip the circuit breaker.
Under the new NEC 2014 rules, you can route the ground wire any way which is practical, back to the panel the breaker is in. The ground wire does not need to run with the conductors. Also, you don't need to home-run all the way back to the panel, if you can reach another circuit whose grounding path goes back to that panel. (And the grounding path must be large enough; if you need to run a 12 AWG ground wire, a path involving a 14 AWG ground wire is not big enough.)
You don't need a separate grounding rod if the building is attached. Water lines are fine, but never use a gas line as a ground rod.