Ground as a conductor
Yes an no.

As a schematic... No. The ground symbol simply means both those wires are the same node and that node is deemed to be your reference ground for measurements. It is an arbitrary and abstract definition of ground. No current ever flows thorough that point to anywhere.

A ground symbol in a schematic is actually only a special net name designator. The following two schematics are equivalent.



As such, what your schematic is internally represented as in any schematic editor is this. Notice GND is a single node/net that isn't actually connected to anywhere outside the circuit.







simulate this circuit &ndash; Schematic created using CircuitLab

Now you may ask what about trace resistance. Well, if you open that can of worms, whether any current flows through R1 would depend on how you routed the ground net.

This way R1, on the right, would carry a small current...



This way, it wouldn't..





If you take your drawing as a "wiring diagram", where each grounding point is a different connection point to either a common ground or physical earth, yes there will be a finite resistance between those points. 

In a real schematic you would indicate separate grounds differently.







simulate this circuit