Why do fuses burn at a specific current?
So if we have a 10A fuse, and it has some arbitrary resistance such as 100 ohms, ...




This typical 10 A fuse has a resistance of 5 m&Omega;. So your guess was out by a factor of about 20,000. At 10 A the power dissipated is given by \$ P = I^2R = 10^2 \times 5m = 500 \ \text {mW} \$.


  RESISTANCE: The resistance of a fuse is usually an
  insignificant part of the total circuit resistance. Since the
  resistance of fractional amperage fuses can be several
  ohms, this fact should be considered when using them
  in low-voltage circuits. Actual values can be obtained by
  contacting Littelfuse. Source: Littlefuse Fuseology Application Guide (which is well worth a read).


The reason for higher resistance in fractional ampere fuses is that the fuse wire is about the same length as the 10 A version but would have to be much finer to blow at, for example, 100 mA. A 100 mA fuse may be protecting a circuit normally drawing, say, 50 mA. If the fuse resistance was 1 &Omega; then there would be a 50 mV drop across it in service.

The required diameter of a fuse wire can be calculated from $$ d = \left( \frac {I_f}{C} \right)^{\frac {2}{3}} $$ where where If is the fusing current in amps, C is Preece’s Coefficient for the particular metal in use. (Source: Ness Engineering.)
From this we can see that a 10 A and 0.1 A (a factor of 100) fuses of the same material would result in the 10 A fuse having a wire diameter \$ 100^{\frac {2}{3}} = 21.5\$ times that of the 0.1 A fuse.


  ... why don't we instead call it 1kV fuse (10 A * 100 ohms), or 10 kW fuse (10 A * 10 A * 100 ohms)?


Because it is an over-current protection device. Fuses already have a voltage rating that means something completely different. See below.

The fuse needs several ratings:


The current (which I think is obvious enough).
The voltage rating of the fuse. This specifies the maximum voltage it can break reliably without forming and sustaining an internal arc.
The time rating - how quickly it will blow.


The Littlefuse article covers all of these in great detail so there is no need to reproduce it here.