Asphalt driveway washed out up to the edge of the asphalt due to heavy rains. Previously put 2 to 4 inch rocks and hard red clay soil at road edge and around top and sides of 36” covert but all of it washed away with the rain and water. How can I repair this myself.
I suspect your problem is not the stone you used, but the fact that the water is draining through and undermining the soil underneath, allowing the rocks to wash away.
I don't know what area of the country you are in, but in the Pacific NW I used to see large rocks encased in wire cages used for drainage. The cages are called gabions and make sure that the drain rock stays in place during floods.
What I would do is this:
Excavate about 1-2 ft around the sides and 3-4 feet out from the end of your culvert.
Place heavy landscape fabric on top of the excavated soil. Secure it with wire staples. The landscape fabric will serve two purposes. First, it will keep the rock from sinking down and mixing with the soil when everything gets wet and soft. Second, it will allow some water to infiltrate so you don't just push all of the flooding somewhere else.
On top of that, build your gabions. You can use cattle panel, chicken wire, or even chain link fencing depending on the size of your drain rock. The gabion can be square, rectangular, etc. The idea is to fully line the culvert area as well as up to the road bed, almost like giant porous bricks.
Fill with your drain rock. I would suggest rip-rap. It's usually inexpensive because it isn't super pretty and the pieces are anywhere from tennis-ball to basketball sized.
Fold the gabion casing up around the sides and over the top of the rock. Secure with wire ties to keep the wire mesh from opening.
If you really want to make sure it stays in place, drive some metal T-posts through the gabion into the ground to lock it in.