In my experience you would want to use "splash".
Throw implies a slightly more aggressive action. For example, spoken in anger, "Why did you throw water on me?" Similarly, someone that says "I went to the bathroom and threw water on my face." is probably trying to project a masculine image.
Splash has a playful connotation. Splashing is what children do when they play in water, e.g. "Sara is splashing Bobby in the pool". It takes on a more neutral connotation when used in other contexts, "I went to the sink and splashed water on my face."
Spill is only used for accidents. To me it implies an accidental "pouring" action where gravity does most of the work. It's hard to imagine someone spilling water on their own face, but it's not impossible either.
Pour is used when a cup is turned upside down and its contents go downward toward the ground. For example, "She poured water on the floor." or "She poured water in the cup." A person can pour water on their own face, but they would have to be holding the container above their heads and willing to make a mess. Athletes are known to do this when they're outside after a workout.