We simply may never know the exact route by which the very earliest or first door closing devices were invented and I suspect the door closer in some way shape or form has been invented and reinvented over and over throughout history.
Arguably not as an important a discovery as fire, when fully considered you may agree however they deserve far more recognition than they have ever previously received and could well be in the top 100 of all time most important mechanical devices ever invented, behind the wheel or the lever but nearly equal of the lock, affecting our daily lives in ways you may have never even thought, achieving the rare position of being practically indispensable and yet to all intents and purpose invisible to all but the most inquisitive of us.
This 'modern' door closer industry can be said to have begun in England on a Tuesday in the summer of 1786, for it was on Tuesday June 13th in 1786, that a linen draper of Cheapside, London, Francis Moore, was awarded the first English patent for a device to close doors, which consisted of weights, a line and pulleys. This was followed four years later by Henry Downers 1790 patent that used a spring to overcome the weight of the door to return it to the closed position.
Patents are of course important they do not however reflect the entire door closer story as not all inventions are patented. Ironmongers were selling door springs up and down the country around the time of Moore and Downers patents were granted.