I think a lot of people already made it clear, but even if the eagle managed to take Frodo and land on mount doom, even if Frodo able to get in the crack of doom all unharmed, he still won't able to throw the ring down. He cannot resist the Ring himself, the Ring is destroyed due to the struggle with Gollum, he bit off Frodo's hand and got the Ring, and then Frodo trying to take the Ring back, resulting in both of them falling down, and Gollum and the Ring sink into the lava.

One could even argue that Sauron can just let them walk in and the Ring still won't getting thrown into the pit because the influence is just that strong.