What if Bilbo kept the Ring with him ...

I don't think that Círdan would have allowed the One Ring to be taken aboard one of the White Ships sailing to the Undying Lands. And the Ring had already been destroyed when Bilbo (and Frodo) sailed into the West. However, if the One Ring had been taken to Valinor and not cast into the Crack of Doom then Sauron would have still retained all of the power that he had recovered over the centuries and he would have conquered Middle-earth. The only way to stop Sauron permanently was to destroy the Master Ring.

"Hell hath no fury like that of the uninvolved." - T. Isabella

The Valar (the Lords of the West) would've refused to take the Ruling Ring. This was already mentioned in the Council of Elrond (book version). The Ring (and Sauron) was a problem for Middle-earth to solve, and the Valar sent what help they felt appropriate (mainly the Wizards).