I am confident Pilgrams on the Mayflower threw darts or knives or ... to pass the time on their way to America.
Anone visiting a "Pub" virtually anywhere in the British Empire since before WWI has been exposed to the bar game of darts.
Virtually every British soldier during WWII was given a set of darts. The British and American soldiers fought, drank, and played side-by-side.
We have a recorded history of Darts in Europe for hundreds of years. Darts and other European passtimes were imported just like most of our forefathers.
But the explosion of Darts as an adult, competitive Sport in America came with Viet Nam from the West and the Beatles (British Explosion) from the East; both starting in the 1960's and extending into the mid 1970's.