The short answer
The phrase is Herbert Spencer's. It does not appear in the first edition of On the Origin of Species in 1859. Darwin took it into the fifth edition in 1869, ten years after publication, at Wallace's urging, first in the heading of a chapter.
Why it misleads
It invites two misreadings at once. That fitness is a property of an animal, rather than a relation between an animal and a place. And that the process is about who is strongest, rather than about who leaves more offspring.
Darwin's own book is careful about this. Natural selection, as he set it out, does not say the process is efficient, kind or fast, and it does not say the fittest survive. It says that small differences that help an organism leave offspring, in the place where it lives, will tend to accumulate.
And “social Darwinism”?
A label Darwin never saw. The doctrine that competition between people, firms or nations is a natural law and ought not to be interfered with was not in use in his lifetime in anything like its modern sense. Most of the thinking behind it is Spencer's, and Spencer had arrived at it before he read the Origin.