Tolkien deliberately did not put this in the published Lord of the Rings, but we can get some hints of the shape of this thought from other writings. In his Letter 156 (to Robert Murray, an unsent draft of 1954) he writes that in Gandalf's telling of his return, "the narrative is urgent, and must not be held up for elaborate discussions involving the whole 'mythological' setting," and that Tolkien "purposely kept all allusions to the highest matters down to mere hints" (that is, matters concerning Eru and the Valar and so on).
In drafting the book, he had the idea that Gandalf would return from early on in the process. Tolkien's first rough notes say: "He refuses the full tale [...] The Balrog was destroyed, and the tower crumbled and stones blocked the door of the secret way. Gandalf was left on the mountain-top. The eagle Gwaihir rescued him. He went then to Lothlórien." (The Treason of Isengard, p430). And in the final 'fair copy' draft, Gandalf's story has "Naked I returned, and naked I lay upon the mountain-top" rather than "Naked I was sent back - for a brief time, until my task is done" in the published version of The Two Towers. So Tolkien does not seem to have Gandalf dying at all in the earliest conception, and only decided very late that he would be sent back rather than returned under his own power - apparently in accordance with the view that exceptions to the rules of mortality require a very high authority.
Our most explicit statement comes later in Letter 156, where Tolkien writes:
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'.
As you note, this makes Gandalf's return an example of direct intervention by Eru. It is something which, in the fictional cosmology, supersedes the power of the Valar. In Unfinished Tales, there is a draft of a meeting of the Valar (Tolkien has a marginal note: "maybe he called upon Eru for counsel?" regarding Manwë) at which they decide who to send as wizards in the first place. In this brief passage there is no recorded dissent from the idea of sending emissaries in general, only some jostling to make sure that enough "interests" are represented - Gandalf/Olórin being apparently the special choice of Manwë and Varda. Saruman/Curumo is privately resentful but the Valar, including Saruman's patron Aulë, do not object.
In The Silmarillion, I think there is only one example where any of the Valar (obviously not counting Melkor) express misgivings about Eru's revealed will, which is this:
Manwë sat long in thought upon Taniquetil, and he sought the counsel of Ilúvatar. And coming then down to Valmar he summoned the Valar to the Ring of Doom, and thither came even Ulmo from the Outer Sea.
Then Manwë said to the Valar: 'This is the counsel of Ilúvatar in my heart: that we should take up again the mastery of Arda, at whatsoever cost, and deliver the Quendi from the shadow of Melkor.' Then Tulkas was glad; but Aulë was grieved, foreboding the hurts of the world that must come of that strife. But the Valar made ready and came forth from Aman in strength of war, resolving to assault the fortresses of Melkor and make an end.
There are disagreements among themselves at other times, such as with Manwë's mercy to Melkor, or Yavanna's plea that results in the creation of the Ents. (Aulë exceeded his authority by creating dwarves, but instantly repented and was forgiven.) But the general pattern is that they do not fully know the mind of Eru, and submit to Manwë who knows it the best. Aulë's sadness does not stop him from full participation in the following war, including forging Melkor's chains.
Manwë's messengers to Númenor before its downfall say "Beware! The will of Eru may not be gainsaid" and it seems that the Valar trust Eru's overall plan despite being ignorant of how it will unfold. In the most striking example of divine intervention, the bending of the World, we hear that "Manwë upon the Mountain called upon Ilúvatar, and for that time the Valar laid down their government of Arda"; there is no dissent from them about the destruction that follows.
Following this pattern, since Eru's return of Gandalf represents his intervention to salvage a plan that the Valar had already formed and was on the brink of failure, it seems unlikely the Valar would disagree. Their reaction might be more like that of Manwë's on learning aspects of Eru's plan that were previously concealed (Quenta Silmarillion chapter 2):
Then it seemed to Manwë that the Song rose once more about him, and he heeded now many things therein that though he had heard them he had not heeded before. And at last the Vision was renewed, but it was not now remote, for he was himself within it, and yet he saw that all was upheld by the hand of Ilúvatar; and the hand entered in, and from it came forth many wonders that had until then been hidden from him in the hearts of the Ainur.
Letter 246 (to Eileen Elgar, 1963) calls Gandalf "an emissary of the
Valar, and virtually their plenipotentiary in accomplishing the plan against Sauron", in describing how Frodo could take Arwen's place on the ship to the West. This is further evidence for the idea that he enjoyed the general support of the Valar, not just of Manwë or Varda specifically, after becoming Gandalf the White. And even if they had some doubts, actually defeating Sauron should surely put those to rest.