Which "God" did Varys supposedly see and hear? I know that the most obvious answer would be R'Hllor since the medium was fire, but I think that begs a greater question: "Can the "Gods" in GoT manifest themselves physically, and if so why haven't they done so rather than using human mediums?"
Here are a relevant couple of quotes by Varys (A Clash of Kings - Chapter 44 "Tyrion"):
"I watched him burn my manly parts in a brazier. The flames turned blue, and I heard a voice answer his call, though I did not understand the words they spoke."
And later on in the same chapter:
"Yet I still dream of that night, my lord. Not of the sorcerer, nor his blade, nor even the way my manhood shriveled as it burned. I dream of that voice. The voice from the flames. Was it a god, a demon, some conjurer's trick?"
Varys himself doesn't know who or what the owner of that voice was. Even if gods do exist in the world of Westeros (which is still not a foregone conclusion), they are not the only supernatural beings in the series. We've seen:
- the Others,
- a shadow monster born out of Melisandre,
- the ghostly inhabitants of the House of the Undying,
- and demonic shadows summoned by Mirri Maz Duur.
So calling what Varys heard that night a God would simply be wild speculation.
If Varys' story is true, and if there was really a supernatural entity (of any kind: god, demon or whatever) answering to the offerings being made in the brazier, it might have been a deity (meaning, an entity worshiped by men and perceived to have divine status) known as the Lady of Spears, since they share the ritual of burning someone's manhood (I don't think that such a ritual could be common practice in other god's rituals or common practice at all in Essos).
This Lady of Spears could be, in turn, just an aspect or a different name for any other of the (real?) gods in the world of GoT, since the true nature of the gods (if any) has not been explained even in George R.R. Martin's detail-rich world.
We Do Not Know
The scene in the show is very similar to that of the books.
Varys: With a hooked blade he sliced me, root and stem, chanting all the while. He burned my parts in a brazier. The flames turned blue and I heard a voice answer his call.
I still dream of that night. Not of the sorcerer, not of his blade. I dream of the voice from the flames. Was it a god? A demon? A conjuror's trick? I don't know.
But the sorcerer called and a voice answered.
And ever since that day, I have hated magic and all those who practice it.
Game of Thrones, Season 3 Episode 4, "And Now His Watch Is Ended"
All we know from that scene is what Varys has told us and he doesn't know what the voice was.
However, we do know that the flames can give visions to people, even those who don't follow the Lord of Light, as we see from Sandor Clegane:
Thoros: We can't tell you. Only the fire can tell you. What do you see?
Sandor: Logs burning.
Thoros: Keep looking. What do you see?
Sandor: Ice. A wall of ice. The Wall.
Thoros: What else?
Sandor: It's where the Wall meets the sea. There's a castle there. There's a mountain. Looks like an arrowhead. The dead are marching past. Thousands of them.
Beric: Do you believe me now, Clegane? Do you believe we're here for a reason?
Game of Thonres, Season 7 Episode 1, "Dragonstone"
As such we can speculate that whatever gave the vision to Sandor and the magic to Melisandre, Thoros and other Red Priests is what called from the flames. As for what that is, well we don't know.
For the second part (the greater question) only speculation can be made at this point. For the first question, we will find out in a few days time (episode 1 of season 6 "The Red Woman") that the deity in question is indeed R'hllor.
!SPOILER WARNING! below this point
Israeli actress Ania Bukstein was recently confirmed for the part of "High Priestess Kinvara".
Seeing her in the recent #2 trailer interacting with Tyrion & Varys inside the throne room of the Great Pyramid of Meereen, pretty much confirms this piece of leaked script we got last year from another actress' (Tehmina Sunny) audition for the same part. And, although this piece of dialogue may eventually have been altered for the final version of the shooting script, the gist cannot but be the same, as it is most essential to the overall storyline.
On a personal note, I 'm quite sad it won't finally be Tehmina Sunny as "Kinvara"...