In that thread, i believe it was postulated that after his death, Gandalf's spirit passed to the Halls of Mandos, from whence he was sent back by the Valar.
Today though, i found the following in letter #156 from Tolkien's Letters :
He was sent by a mere prudent plan of the angelic Valar or governors; but Authority had taken up this plan and enlarged it, at the moment of its failure. 'Naked I was sent back – for a brief time, until my task is done'. Sent back by whom, and whence? Not by the 'gods' whose business is only with this embodied world and its time; for he passed 'out of thought and time'. Naked is alas! unclear. It was meant just literally, 'unclothed like a child' (not discarnate), and so ready to receive the white robes of the highest. Galadriel's power is not divine, and his healing in Lórien is meant to be no more than physical healing and refreshment.
So it seems that the Valar had nothing to do whatsoever with Gandalf's return/rebirth.
Indeed, Gandalf passed out of 'thought and time', which would mean that he passed beyond the confines of Arda. It seems that Eru Ilúvatar himself passed Gandalf's spirit to his Halls, upgraded his power and sent him back to Middle-earth to finish his task as Gandalf the White :
Gandalf really 'died', and was changed: for that seems to me the only real cheating, to represent anything that can be called 'death' as making no difference. 'I am G. the White, who has returned from death'. Probably he should rather have said to Wormtongue: 'I have not passed through death (not 'fire and flood') to bandy crooked words with a serving-man'.
(from the same letter)
This is an important point, for it seems that for the second time in the history of Arda, Ilúvatar interfered directly in the course of events - the first time, he took Valinor from the circles of the world and he sank Númenor; the second time, he sent Gandalf the White to Middle-earth.