This question is brought up a lot.
The LOTR Wiki Says
"It has been pointed out by several observers that the entire War of the Ring could have been over almost as quickly as it began if only Gandalf had requested that Gwaihir simply take the ring himself and delivered it to Mount Doom to have it destroyed. Whether or not this is a reasonable possibility is debatable, however, using logic, you can conclude that Sauron would have seen the ring and sent armies and the Nazgûl after him, thus eventually overpowering Gwaihir and ultimately giving Sauron the ring. This also is debatable because it is unknown how the wearing of the Ring on a claw or other appendage would affect an animal and Sauron's ability to see it with his Eye. It should also be understood that the mission to bring the One Ring to Mordor was appointed to Frodo by the Council of Elrond. Only Frodo alone should risk holding the Ring, thus it was not Gwaihir's mission, nor was it his burden, to carry either the Ring or the Ring Bearer. This is addressed in the video game Lord of the Rings: War In The North, when this is suggested by the party to Gandalf. Also, since a single poison arrow had nearly brought him down in the past, it would be reasonable to assume Gwaihir and the other eagles would be reluctant to fly over orc archers."
Personally, I think that Gandalfs Relationship with Gwaihir was one of mutual respect and more about helping each other in a pinch rather than engaging in a taxi service.