According to this British Medical Journal (BMJ) article the white lab coat was adopted in the mid 19th century in order to "restore credibility to a profession that had been damaged by scientists, who had proved that many doctors’ remedies were worthless."

Reference: The Return of the White Lab Coat by Oliver Ellis (September 2010)

Interestingly many health services have recently started to abandoned the white lab coat since it can harbour and transfer bacteria very efficiently.

Reference: https://dx.doi.org/10.7860%2FJCDR%2F2012%2F4286.2364