Uh, no

Doing some research for a story, I searched “what is the best water-cooled machine gun.” First answer from the first web site, quora dot com, from a “former Army Winter Warfare instructor for 35 years 1978-2016: “The Lewis gun.” 

The machine gun invented by Army Col. Isaac Newton Lewis in 1911 was a fine gun, used in both world wars, the Korean War and in Vietnam and every small war in between.

But, the Lewis gun is not a water-cooled gun. It is cooled by the surrounding air.