Okay guys, this is gonna sound a little off the wall, but I'm kinda qualified to tell you all how this fight would go. A: I was once the world online virtual dogfighting champion. I learned my skills from actual top gun trainers in the USAF. B: as luck would have it, I've actually witnessed a small hawk engage a bald eagle in a drawn out battle in the air.
The Osprey, being the larger of the two birds, is the more powerful climber. It is also the less maneuverable of the two. The Falcon, being smaller, can dive faster and here's the part I was surprised to learn from watching first hand: Much like a German Messerschmidt from WW2, the bird that dives quickly can "zoom" back up just as quickly.
What develops is a classic "boom and zoom" fight where the Falcon makes quick dives from an advantageous position and zooms back out of reach again. The Osprey, however, will use it's larger wingspan to steal some altitude after each strike. This is called energy fighting and every dogfighter pilot learns about energy fighting. Birds, naturally, are instinctual energy fighters. So long as the Falcon doesn't get a crippling strike against the Osprey, then each "boom and zoom" attack becomes less and less effective as the Osprey closes the energy gap between the two birds. At some point, the Falcon will make one last, desperate strike, then dive away never to return, it's energy advantage lost.
One mistake by the Falcon and the Osprey will have it for lunch. One mistake by the Osprey and it will find itself in trouble, unable to win the energy fight, unable to escape unless there is a tree nearby to take refuge in. The one thing the Falcon doesn't want to do is enter the branch zone of a tree against an Osprey where the Falcon loses its speed and mobility advantage and the Osprey's size wins out. So if the Falcon gets a crippling strike on the Osprey, the seahawk will make for the branches of a tree. If there is no where to run to, the Osprey will have to take the fight low to the ground and hope for the best.