High-current monitoring circuit
You can use a current sense ampflifier that has an external sense resistor. TI offers several, for example. Then you can choose the sense resistor value as low as you like to suit the current levels you need to measure.

With high current, for accurate measurements you'll want to be sure that your sense resistor has a low TCR so that self-heating doesn't cause the circuit's overall gain to shift.

EDIT 

Looking at the datasheet for the part you mentioned, you are already using the correct type of device. The normal application circuit for your part looks like this:



The Icc spec is a promise by Maxim that the current going in to the VCC pin of this chip will not exceed 2.2 mA. It has nothing to do with the current being sensed, which is indicated by the ILOAD arrow on the diagram.