Current is measured in amperes, so no. High impedance takes more voltage, less current, same power if the sensitivity stays the same.
I'm gonna throw some numbers at you, get ready to duck. They're equations to relate power to voltage and current.
To find power (in watts, labeled P) from voltage (in volts, labeled V) and current (in amperes (I think whole amperes), labeled I for some reason), you do this:
P = I x V
To find power from voltage and impedance (R), you do this:
P = V^2 / R
To find power from current and impedance, you do this:
P = I^2 x R
So as might be able to see, as impedance increases, voltage must increase to achieve the same amount of power, and current must decrease to achieve the same amount of power. Because it's the headphone's sensitivity that determines how loud it gets with a given amount of power and not the impedance, as impedance increases and sensitivity stays the same voltage must increase, and current must decrease.
I can throw numbers in there so it looks less alien, if you need it.
this is true if you think about electricity as electrons.
the coil is only willing to accept a certain amount of electrons.
if you group those electrons into large groups, they are called amperes.
if you group those electrons into small groups, they are called voltage.
the above post is intuitive to say 'sensitivity and RMS value determine how many electrons can be transferred'
how many of those need to be in small groups is determined by the ohm rating.