By the power of truth, I, while living, have conquered the universe.
No, Halo 3 provided absolutely no closure at all. In fact, the entire final mission is just a set-up for Halo 4.
Let's start out with the final Terminal, the first thing we see. Mendicant Bias tells John that he's sending him to the Forerunners as proof of his atonement, this alone sets up the premise for an entire new trilogy and providing the opportunity to delve into the story of the Forerunners.
Then the Gravemind. He somehow manages to survive the destruction of High Charity and begins rebuilding himself on Installation 04B, then when the Ring begins to rip itself apart he says that the destruction of the Ring will not be the end of him - it will merely delay his return.
Then there's the Covenant. We killed Truth and they fragmented, yes. But it's established in the description of the Assembly map that: "The Covenant war machine continues its march to conquest; even with its head severed it is still dangerous" The 'head' clearly refers to Truth and, by extension, the San'Shyuum, who have mysteriously fled to an unknown location. This suggests that the Covenant may be crippled but still has the potential to return - and we know that they do in Halo 4 at the opening of the game.
Then the Flood. Contained on the other Halos, but we don't know exactly what is transpiring on these other Halos. Penitent Tangent went rampant, neglecting his duties and therefore allowing the Flood to escape containment - we've seen how Monitors can become privy to rampancy after so much time alone, therefore the same could well happen on other Halos leading to more Flood outbreaks. And as a result, catastrophic ramifications, as speculated by 343 GS in CEA about what would happen if the Flood took control of a Halo.
Speaking of Halo, there are still 6 other Rings primed and ready to fire. As long as these Rings still exist, that's going to be a shadow constantly looming over the the galaxy.
Also, there's Cortana. What kind of closure would it be if she's just left standing there on the Dawn adrift in space for the rest of eternity? Is that really the ending that she deserves after we went through 3 whole games with her, the latter of which we spent ages trying to rescue her. Would that really be what you call 'closure' and 'satisfactory'?
To sum this all up, Halo 3 left so many loose ends it just wasn't even funny. There was no sense of closure, other than this idea that the story for John had come 'full circle' by sticking him back in cryo to reflect the opening of Halo CE. Yet they still added in the fact that he's been sent to the Forerunners which is a complete contradiction to the closure they were seemingly trying to implement...
Frankly, anyone who says that Halo 3 brought closure to the series is talking a load of twaddle!
[Edited on 05.02.2012 1:43 PM PDT]