Why is an aircraft carrier's arresting cable made from steel?
The single most important thing for an arrestor cable is that it works reliably, compared to this incremental differences in weight or durability are fairly minor considerations. 

One of the major advantages of kevlar is that it has a very high strength to weight ratio, however all this really achieves in this context is that the cable is thinner and lighter and I would guess that the weight of spare arrestor cables is vanishingly trivial in the context of the total payload of an aircraft carrier. 

Another consideration is that Kevlar has much lower stiffness than steel, which will be further exacerbated by the differences between a fibre rope and a braided steel cable. This means that it will extend further under the same load. Not only does this mean that it allows the aircraft to travel further before it stops but you also need to fundamentally redesign the whole arrestor system to account for the very different material properties of the wire and then the new design would need to be tested, evaluated and certified. 

Also, while kevlar is considered abrasion resistant by the standards of textiles it is nowhere near as resistant to abrasion as high strength steel cable. Consider that industrial steel cables may have a design life in the order of decades (think of suspension bridges) whereas high strength polymer climbing ropes are typically retired after a few years, mostly because of the cumulative effects  of abrasion, dirt and UV exposure. Also steel is almost entirely unaffected by contamination by spilled fuel, solvents, lubricants, hydraulic fluid etc. 

Similarly the arrestor hooks on aircraft are made form high strength steel, which is much harder than Kevlar so even a small burr or sharp edge (as you miht get form landing or combat damage) could cut through a kevler rope like knife whereas a steel cable is much more resistant to sharp edged shearing. 

125 loading cycles is nowhere near where metal fatigue would be an issue, especially in steel, so it seem more likely that the life of the cable is based on direct wear and local overloading from the arrestor hook. Steel being relatively hard while having reasonable ductility  is much better at tolerating this sort of wear and tear than most other materials. 

It's also worth noting that overloading beyond the yield stress (but before total failure)  will often produce  a noticeable kink in a steel cable whcih is a warning sign that it shod be replaced. This gradual failure mode (ie damage is measurable before it becomes critical( is one of the major reasons why steel is preferred for safety critical structures ie you get some waring of impending failure. 

However the most compelling answer is that steel cables are proven to work well for the job and so there is no compelling reason to redesign the whole system to use a new material for the sake of fairly marginal (if any) benefits in performance. 

It's also worth noting that textiles are use for the arrestor nets for emergency landings where it is known or suspected that the landing gear on a particular aircraft is compromised. Here the light weight and greater stretch is a distinct advantage as it mitigates potential damage to the air-frame.