Depends. If he killed a child, people would have cared. If he kidnapped someone and tortured them to death, people would have cared. If he shot up a theater, people would have cared...etc. Meanwhile there are plenty of other rich American trophy hunters going around doing shit like this all the time that doesn't command internet pile-ons. We can't always tell what will or won't go viral with these things.

But I think overall, people generally have more readily deployed outrage when there is a blatant disparity in power on display, specifically when there is a level of innocence involved. If he had shot this lion while it was charging to attack him, it wouldn't have become a huge story like this. Two adults having a dispute where one shoots the other one. We assume a level of agency for both and (right or wrong) don't think much of it. (although people do certainly care about the bigger picture of violence and murder and specific incidents will jump out and catch national attention from time to time just the same)

Since animals don't have that same degree of agency we have, people tend to be more sensitive about it in general. That there is a large degree of shadiness surrounding the actual legality of this particular incident(not to mention a level of cruelty) and that there's as yet little evidence to suggest this guy is a responsible, conservationist, anti-poaching, respectful sort of hunter who does everything right and some evidence to suggest he's basically just a dumb asshole who wants to have his fun at the expense of others, well it's not hard to see how this can spiral into such a frenzy. Combine the Internet overall and that frenzy quickly becomes a bit nauseating regardless.