If it is at all possible to strip out emotion and look at fundamentals: Yes, life on earth will go on with or without us someday. Sure, even that life will be doomed in the distant future. Humans are a species that are a byproduct of the environment just like every other species and we are dependent on the system for prolonged survival as a species. The rules haven't changed. New animals migrate into an area, or else some subset evolves beneficial traits for survival and other animals go extent as a result. Nature made us what we are, and what we are is causing extinctions.
However, we are different because our effect on the system is greater then then any other species the system has ever encountered. If we can not strictly say that current extinctions are non-natural, we can say that they are non-normal. We are throwing the whole system out of balance. It will regain balance one way or another regardless of our actions. That balance may be an ecology of humans, rats, pigeons and cockroaches only, or it could be a world with no humans and rich biodiversity. The curious thing about the current system is that we, as the major predator, can understand the system (to a degree) and modify our actions to effect the eventual balance point (to a degree.) We have the potential to mitigate our effect on our environment.
This potential is why extinctions are a moral issue. We are choosing to do things that we know negatively effect life around us, be it human or animal. From here we can endlessly argue morality, but we can't shrug of responsibility. We can not wright off species because they did not evolve quick enough when we make decisions that modify their environment faster then process of evolution can cope with. We can not ignore an extinction because they would have died out tomorrow, when our choices made them die out today. Above all we can not abdicate our responsibility just because we ourselves are products of the ecology. It is the traits that evolution bred into us that have allowed us to dominate the environment, and it is those same traits that allow us to mitigate our own destructive tendencies.
Ultimately I see my own hypocrisy. My standard of living is the product of a lot of environmental destruction. But, I won't excuse myself ether. We are diminished when a spiecies goes extinct and we have only ourselves to blame.