We do actually speak of Newtonian mechanics, for what it's worth. Probably more than anyone in science actually speaks of Darwinian evolution. So we've sort of already done what this guy is asking for, it seems?
Why Newtonian mechanics? Simple:
If it is very small: use quantum mechanics
If there are a lot of little things to describe: use statistical mechanics
If it travels very fast: use relativistic mechanics
And so on.
The non-quantum, non-statistical, non-relativistic, etc... version was the first to be formally described. Calling it Newtonian is the simple way of distinguishing it from the more complex variants. Among physicists, you would refer to classical mechanics, which also refers to its historical status.