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 mΩ. 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 Ω 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 I_f 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.

1. The conductor fails when it reaches a certain temperature. Because the fuse is in thermal contact with ambient, it can dissipate a certain amount of power before it blows.
2. Your 10A fuse is designed to blow at 10A (plus or minus some tolerance). So it should run at 9A all day.
   • But that 10A fuse will take a good long time to blow at 10A, and will blow much quicker at 20A, and may misbehave if you shove 100A through it. There's a whole, mostly neglected science to fuses.
   • And if you run that 10A fuse at 9A or 9.8A all day, it'll run hot and slowly degrade.
   • Which all means that if it really matters how fast it blows, or how long it lasts, that you need to talk to the fuse manufacturer.
3. Fuses are rated in amps because that's what most folks installing fuses care about. The ideal 10A fuse drops no voltage, and neither blows nor degrades at even a nanoamp below 10A, but blows immediately (or after a well-defined time) above that. No ideal fuses exist.
4. While you're pondering all of this, you may want to dig up some fuse data sheets and look. The good companies (Bussman, Littlefuse, etc.) specify this -- and there are such things as slow-blow fuses that are designed for temporary overload, and fast-blow fuses that are designed to react more quickly than "ordinary" fuses. If the way the fuse needs to respond is non-standard and critical, it can become quite the engineering exercise to design one in.

Normally, a fuse doesn't know what voltage circuit it is used in - it only knows the current that is flowing through it, so that is the only thing that can cause it to blow.

Fuses also have a voltage rating because, once the fuse blows, it will have the full circuit voltage across it, so it must be designed to safely handle that voltage without arcing.