Um, no. Humans do have morality, and can and should apply it to one another - and can expect and even demand the same in return from others. We can't do that with nature, and nature does not need or want that in return from us.
Humans are both animals, that need to fit into an ecosystem, and social creatures, that need to fit into a society. Both domains require rules for how we interact with and fit into them, but those rules are not the same.
I think the only flaw there is that our morality came from nature and therefore is one of the things we use to fit into our ecosystem. There's nothing to say we shouldn't apply it to other members of the system. If nature didn't "want" that we wouldn't be capable of doing it.
Given how far up the food chain we are it could be argued that our morality exists precisely for this purpose.
What on earth makes you think that? No, I don't think humans are super-natural. But in addition to being creatures sprung from nature, we are also something else (indeed, the very definition of something being anything other than natural, i.e. cultural or artificial, is that humans have had a hand in its creation).
Humans are both animals, that need to fit into an ecosystem, and social creatures, that need to fit into a society. Both domains require rules for how we interact with and fit into them, but those rules are not the same.