MOSFET driver without gate resistor
The chip specifies the "typical" LO output current as 1A, not an absolute maximum. This current is limited by its internal output resistance since there doesn't seem to be a limiting circuit according to the datasheet's diagram of the output stage. It won't be able to deliver 1A (or more) continuously, however - it can only do that for short pulses. So if you just short its output to ground, it will die.

MOSFETs like the ones in the MIC4605's output stage can handle large current pulses for a short time by simply absorbing the heat generated by the pulse in their thermal mass. It's not current that kills a FET, it's overheating. If the pulse is short enough, the FET will not heat up much and therefore survive just fine.

Take a look at the datasheet of the IRLML2803 MOSFET, for example. On page 4, bottom right, there's a graph showing the maximum safe operating area (SOA) of the device. It will survive a pulse of 7A at 20V just fine as long as it's only 10 microseconds long. That's 140W dissipated in a SOT-23 device. The FET can't get rid of that heat but will absorb it in its thermal mass instead, heating up a little. As long as that doesn't cause it to get hotter than its maximum junction temperature, no harm will be done.