Probably is related.
Apparently not, according to Prof. Google.
Rugged individualism definition:
The belief that all individuals, or nearly all individuals, can succeed on their own and that government help for people should be minimal. The phrase is often associated with policies of the Republican party and was widely used by the Republican president Herbert Hoover. The phrase was later used in scorn by the Democratic presidents Franklin D. Roosevelt and Harry S. Truman to refer to the disasters of Hoover's administration, during which the stock market Crash of 1929 occurred and the Great Depression began.
Social Darwinism definition:
The theory that individuals, groups, and peoples are subject to the same Darwinian laws of natural selection as plants and animals. Now largely discredited, social Darwinism was advocated by Herbert Spencer and others in the late 19th and early 20th centuries and was used to justify political conservatism, imperialism, and racism and to discourage intervention and reform.
The latter has been discredited for the reasons mentioned -- I think it was also used to justify eugenics -- whereas the former concept is still widely embraced.