Water enters at the right and exits at the left

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Soil/water exits at this end. I busted up the asphalt.

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This is the roof valley that I believe the water is coming from.

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This is the area it started. The eroded channel under the driveway is about 6 inches in diameter.

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jmcc....remove the asphalt. Fix the downspouts! Backfill the erosion with drainable material. It looks like the material is a slightly clayey, silty fine sand. You could also add an underdrain; however, not really necessary if you fix the erosion source.

Extend a concrete apron from the garage outward for a few feet, the you can use asphalt....or concrete the driveway.

Along with Ron's thought the material you fill with should be a filter to hold back the fines yet let water pass. An all 'round filter is concrete sand. If in doubt look up ASTM C-33 fine aggregate. However, at the discharge end you need to hold the concrete sand some how. I'd place a layer or "block" of porous gravel, say #4 size (1/4")up to an inch or a little more. Pack the concrete sand in to fill any cavities that do not show. Patch the top area with blacktop or concrete. An alternative that "may" do the job is pack in the concrete sand from both ends and use the gravel as final fill on he outlet zone.

I had a similar problem with a concrete driveway. I cored holes in the slab and forced in (pounding in one little bit at a time down the cored hole 3" diam. with a 2-3/4" diam wooden ram )dirty sand containing a small amount of bentonite clay (4 percent) and a little cement 6% in a sloppy slurry.. It took several such cored places to finally stop the erosion. Soil here is fine sand. The loss of mix water in that sand stiffened up the mix and so only small areas of voids could be treated, meaning plenty of cores needed. Ended up with them on about 2 ft centers. The cement, bentonite and sand in the mix made the fill sand very impermeable. This also could work at the subject site, but then will a new path develop? In my case there was an alternative path for the water. If you use bentonite, go easy because it likes water and can expand GREATLY.

With no rain gutters it is not surprising that this would happen. Water has to go somewhere, and the soil below the driveway was fill.

Urethane injection could work (if you plug the open ends), but the job may be too small for many contractors. The price has come down and there are more contractors doing residential work, so it may be an option. On the other hand, with the damage to the asphalt it would be to cut out the asphalt, deal with the soil and drainage, and have that area repaved(or extend the concrete apron).

It is a driveway and garage entrance.
Have you taken steps to prevent concrete collapse/cracking by the loads before the repairs are complete?