To your question, current is measured inline with the load, and it is not done with the voltage part of the meter. However, as soon as you hook up a meter inline with the load, to check for current flow, the path is completed, (if it did not already exist) and there will be current flow, even if there was not to begin with, and voltage will drop to zero. (The potential of the neutral, and equipment grounding, conductors.)
There are inductive ammeters that do not connect inline with the load, however, you can't check for current flow in the tank, with one of these, and if there is no path to ground in the tank, (a grounding probe) there is not going to be any current flow in the tank, even with an impressed voltage of 120VAC to ground. (an exposed hot wire.)
In salt water, there will always be some minute current flow, as it is an electrolyte (think battery) coupled with inductive loads, the voltage can build to 12 VAC and above. Such small voltages, even with current flow, are of no concern. A tank running, may display a zero voltage potential, and a small current flow through the metal main shaft of an external pump. Not much can be done about that.
The only time you should be concerned is if you stick your hand in the tank and you get bit. Even that does not indicate a problem, but don't stick your hand in a second time to see if you really got bit or not.
Tank water to common ground with the pumps power source, can show a voltage potential difference if one exists. Whether it is induced or impressed is a bit harder to evaluate. The main source of a voltage, with sufficient magnitude that could cause harm given a path to ground, would be from a high resistance short in a line cord, (or a broken heater,) and the way to check for that is with a mega meter. If it were a low resistance short, it would either trip the GFCI, or the main breaker, depending on where and how it was shorting.
At the same time, these types of shorts will not cause general current flow in the tank (unless from the hot of one cord to the ground/neutral of another) and any current flow there would be, would be local to the problem area, taking the path of least resistance.
Simply, if you are not having an issue, does not do much good to go looking for one.