First off, gandalf did not know it to be the one ring. Bilbo was quite protective and mysterious about it. And such a powerful thing does not randomly pop up in the hands of a hobbit ( or hobbits if you consider the fact that he stole it from gollum, who was a hobbit). That being said the eagles were their own race of intelligent beings. Gandalf did not command them he would have to ask their help. They were on the other side of the misty mountains (if you read the hobbit you would know that they are encontered by the dwarfs and bilbo on the other side of the misty mountains after bilbo had found the ring). Gandalf also wasnt all knowing or all powerful, in fact his duty in the realm of middle earth was to inspire men to reclaim their greatness. He was the holder of the red ring of power that would help put fire into the hearts of men. He got the ring from Cirdan the shipwright, who at the end of the books takes them all across. The Prime God of that universe was very particular about the lesser gods interferring with humans. The elves were allowed to be meddled with but even the lesser gods did not know what became of humans after death. The dwarves came about because the lesser god of the earth got anxious for the elves and made his own race out of stone. These were decided to be okay and could be monkeyed with, but humans were off limits.
That being said Sauron was a corrupt being that had tremendous power much greater than anyone. The lesser gods were above him, but he was part of the beings directly below them. He was corrupted by the evil lesser god and was a necromancer and a huge threat. Gandalf would not have a chance. The bulrog he fights in the misty mountains was well below Sauron in power and Gandalf died in the fight. A eagle would have no chance if it was detected. The eagles went through to save sam and frodo because Sauron's fleeting might was concentrated on distroying the line of kings, what he assumed to be the real threat. It was only luck that sam and frodo were able to destroy the ring.