After the fight with the Balrog Gandalf says to his fellows "Naked I was sent back - for a brief time, until my task is done." (The two towers, ch. The white rider).
The three rings (and the one ring) are, albeit magical and special, physical objects: which can, as we know, be physically destroyed.
As far as I understand Gandalf already at the time of meeting the Balrog was the ring bearer of Narya. So, why doesn't he lose his ring before his "resurrection"? Since he comes back "naked" it would seem natural that all his physical possessions are gone with the Balrog.