How important is it to make eye contact with your partner?
Sometimes, not as important as you think.
Watch this vid, but especially starting at about the 1:45 mark.
You'd never know watching the film, it's that frikkin' smooth. And they're that good.
What looks natural to the camera, and what looks natural to you the actor may be two entirely different things. And you still have to make it work. It's what they're paying you for.
I'm not saying you're acting all by yourself up there. You're not. You've got to respond to your partner(s), the scene, the weather, and a whole host of other things that the character would logically be amidst of.
Sometimes though, it's as much, of not more about "the shot" than it is about you.