I like the wood look...coordinates well with your hardwood floors:)

I normally say no, don't paint wood, but in this case I'm going to say yes. You have a nice view and everything else is painted. So the wood stands out. I think if you paint it, the focus would go more quickly to the view and less on the wood. Try taping on paper over the wood and see if that's true. Does the view stand out more with a painted door?