Actually it is the power (volts times amps). So a more powerful atty will have a lower resistance; the resistance determines how much current will flow for a given voltage.
Of course, there is still a limitation. Half the length of a standard coil will have half the resistance and twice the current will flow; but it will get too hot and burn out.
A thicker nichrome wire would handle more power ok but it would have both a greater heat capacity in itself and a greater surface area for heat transfer (loss); hence more power would be required to reach the same temperature than a simple scaling of resistance would suggest (it is not a linear relationship). But using a slightly bigger diameter wire with a slightly lower resistance could be a good idea; so long as the battery can put out sufficient current.
A better battery is better because it has a low internal resistance - can give more current; in other words the voltage is less divided between the atty and itself (less voltage drop).
Fuses are rated in amps because mains voltage is a constant.
So 4 volts 1.5 amps would give the same performance as a 6v 1 amp device.