It depends on the return ground or on the question whether you complete the circuit, using the words from the question.
The confusing part in your picture is the A.C. generator: It is shown without its two terminals and surrounded by the equipotential waters of the Atlantic Ocean. Therefore, all the current it creates will stay right in the Ocean. Another potentially (pun intended) misleading detail of your question is the earth/ground symbol on Mars, and I suspect this is the root of your troubles. Thus, ...
I have taken the freedom to rearrange your setup, just using another picture for the generator, with two terminals. One end of the generator is connected to Earth (ⴲ), the other one goes to a light bulb and from there to Mars (♂), complete with the copper rod you mention. Note that the light bulb remains unlit ("off").
If you want current to flow and the light bulb to be on, you need a return wire, closing the circuit like this:
For the circuit to work, you don't need anything to be grounded or earthed, what you need is a whole circuit where the current can flow in a loop. For the fun of it, let's use the earth/ground symbol arbitrarily and create one circuit that's Earthed and one that's Marsed. Note that this is just an additional symbol used at our discretion to obtain a common reference for the entire thing - it is not something we need for the circuit to work.
What we really needed for the circuit to work was the return wire from Mars to Earth. This return wire actually puts Mars and Earth at the same potential, so we can even consider Earth to be marsed and Mars to be earthed - this is still the same arrangement, no matter if we use no, one or two Earth/Ground symbols:
Now, there may be one last question left: What happens if we use a Ground symbol in the first picture, the one without the return wire? Like this:
Well, the bulb stays off. Again: It is not the symbol, it is the return wire that's needed, and we don't have it in either of the last two pictures.
Hmmm... If the circuit doesn't really care about the Ground/Earth symbol, why is it so commonly used by so many? What's the point?
Check this out:
The light bulb is on, but the return link is missing. This is possible because, using ground symbol magic, all parts of a circuit diagram that are attached to a ground symbol are considered to be connected, even if no one took the time to actually draw the wires that connect all the grounded bits of a circuit.
These are all the same, just using a different notation:
The ground/earth symbol is so handy that we use it often, and when we are sloppy, we sometimes don't even pay attention to properly calling it ground or earth. Strictly speaking, an earthed circuit is connected to Earth or safety ground (literally: a big rod next to your house or garden, driven into earth ground), and a grounded circuit can be connected to the (-) end of a battery driven device, with no actual connection to the real ground. This is confusing, but it's jargon and everyone uses it.
Armed with all this knowledge, you can figure out why this is plenty funny (image source)...
Note (1): I have used the astronomic symbols (ⴲ, ♂) for
a reason. In doing so, I was free to use the electrical Earth symbol
on either planet.
Note (2): The A.C. generator could also be a D.C. battery. For the
purpose of understanding this setup, you can think of an A.C. source
just like you would of a battery being reversed 50 or 60 times a
second (depending on national standards). To explain why D.C. would
not work well on earth and in the ground, with no physical return
wire, made of metal, consider electrolysis and moisture in the
ground, to start with... This is a whole 'nother story...