If you get rid of the swag, you could (in theory, I can't see what's to the right of the door) have a curtain on the door too; however, right now the operational side of the door is on the wrong side to make that practical.

Actually, that curtain is a touch too formal for your furniture, I think. Keep it simple!

If the window had been dressed with very simple draperies, then I suppose you could have done the same with your French doors. That the window treatment is a formal style, replete with swags, cascades, side panels and tie back, it is a complete picture onto itself. Therefore, I personally see no need to "fashion" the French doors. Often times this has to do with personal style. To my eye, everything is quite complete.