Is there any connection between Darwin's views and Sherlock Holmes stories by Arthur Conan Doyle?

This is a really interesting question!
Holmes scholars, you may have a better idea here, but I'll
give it a shot.

Darwin lived from 1809-1882 and Arthur
Conan Doyle lived from 1859-1930.  Darwin was a naturalist and Doyle studied medicine
and eventually practiced opthamology. Given that they were partly contemporaries and
that they were both living in the same country and that Darwin's ideas were
revolutionary at the time (and still controversial for some today) and that both men
were involved in scientific/biological careers, I would say it is safe to say that Doyle
was certainly aware of Darwin, likely influenced by
him. 

He quotes Darwin in "The Study of Scarlet" (I think
this is the right story):

"Do you know what Darwin says
about music? He claims that the power of producing and appreciating it existed among the
human race long before the power of speech was arrived at. Perhaps that is why we are so
subtly influenced by it." 

You could easily make the case
that there is a general parallel between Darwin's gathering of information and Holmes'
gathering clues. You might say that Darwin's inductive process incorporated elements of
deduction along the way.  In other words, like Holmes at times, Darwin began to
inductively piece together clues and the idea of evolution and species adaptation began
to develop. Then, he deductively began to organize and gather data to support that
hypothesis.  Holmes scholars would do better than I on this: But Holmes would recognize
specific, detailed clues and inductively form conclusions; then, he would have an "aha!"
moment and would then have the overall conclusion; then he would deductively put all the
pieces together. I don't really think he got this idea from Darwin; it is just part of
the scientific process.

Lastly, in terms of
religion/science, Doyle was a Christian Spiritualist so his ideas of reality would
probably contrast pretty starkly with Darwin's more scientifically based world.  Darwin
pretty much referred to himself as agnostic.  Both extremely intelligent guys, so who
knows; Doyle could have been as open to the most progressive scientific ideas as he was
to Spiritualism. Check more bio on Doyle, inductively pick up clues; form a hypothesis. 
Then, deductively find the clues and organize the data to support your hypothesis.