The picnic was canceled, due to rain.

Correct: "The picnic was canceled, because of rain" or "Cancellation of the picnic was due to rain" or "The cancellation, due to rain, was a problem for ...." The word "due" is an adjective, a noun modifier. It is allowed to modify a noun, as it does in the second and third structure here. It cannot modify the action of a verb, as the original has. For that, we need "because of," which opens a preposition phrase that can modify verbs.