Without seeing, I cannot say for sure. The real secret to a window which will not leak is flashing. Stucco is not a water barrier. You have to assume water will be behind the stucco, running down hill. The proper installation has flashing ( building paper is OK ) flashed down over a flange on the window. From top to bottom, every piece of flashing overlaps on top of the piece below, so water never has any crack where gravity can take it.
Stuffing a windo with enough caulk may achieve success, but again, I cannot tell from your description if the window frame is actually properly flashed .