All the white onions have been used for something else, all I have is the red onions, I would cook the red onions the same way, according to the directions, correct?
While I've never seen it done, a quick review of several popular French Onion Soup Recipes either do not specify the type of onion or specifically allow for it. So the answer to your question would be 'yes you can'. The food police will not come busting through your door for making "French Onion Soup" with red onions.
Now, the next question is "Will anybody like the results?"
You should probably start with a simple substitution of the red onions and follow the same recipe. (The linked recipe doesn't distinguish between red or yellow onions in amounts or process)