I believe it would also be classified as murder if he chased her and she fell resulting in her death.
I think so too. Federal statutes say something about "prior similar conduct" as a factor in classifying something as 2nd degree murder (at least).
The only way I can see that this could be a lesser charge would be something like he fixed a rope for climbing, did it improperly, and she fell to her death. However, the burial/body concealment part absolutely makes it a greater crime.
Would the coroner have put that as "accident" or "undetermined" and not homicide?
I don't think so. I think the reason they say "homicide" is that upon gross examination of the body, it was obviously homicide. My guess is that she had major trauma somewhere that left clear evidence in the bones. And if not major trauma, then minor bone trauma consistent with strangulation.
The reason I think it might be strangulation (or a head injury) is that he seems to have wanted her to "shut up" (and vice versa). He strikes me as someone who is used to a lot of quiet time (those rock sculpture skills, all that sketching and design work, the amazing paintings he had at home) and now he's being "directed" and photographed daily. I do wonder if they had a daily plan - to get to X place and take Y picture, the way so many youtubers and instagrammers do. We've been yelled at so many times in museums and in parks, because we were actually
not taking a picture, just standing there enjoying something that we had come to see. It's changed how we travel. The General Sherman tree in Sequoia has to be visited after 4 pm (the park practically empties at that point - because so many drive in for a few hours, snap that one tree, get in the car, drive to Kings Canyon (two snapshot spots heavily impacted). Yosemite is the same way, although no one expects a "clean" photo of just themselves in Yosemite (most of the time). We ourselves now do dispersed (NF not BLM) camping. BLM camping is way too scary for me. Everyone seems to have a gun, lots of solo people who have been out a long long time (in California, these are very desert-like places and quite deserted, so if someone does camp nearby, it's awkward).
National Park camping is much more supervised in the major parks (like Grand Teton). National Forests, less so, which is where Gabby was found. BLM has no supervision, IME.