An early written example of the term is in an 1850's cookbook, Cookery for Maids of All Work:

French Fried Potatoes.—Cut new potatoes in thin slices, put them in boiling fat, and a little salt; fry both sides of a light golden brown colour; drain dry from fat, and serve hot.

The shortened term french fries seems to have originated in North America, and is in use by the late 1800s. Here it is used in Good Housekeeping in 1899:

The perfection of French fries is due chiefly to the fact that plenty of fat is used, and that the thing fried is wholly immersed.

So no, the name does not come from confused American soldiers (presumably the folk etymology is referring to soldiers in one of the world wars). There are conflicting claims that fries originated in Belgium or France, and references to fried potatoes in both countries go back centuries. So this story may have been created by someone promoting the Belgian claim in order to explain away the name used in English.