Does the Giant Impact Hypothesis explain how the Moon circularized its orbit? I just read most of Wikipedia's article on the Giant Impact Hypothesis. Basically a large object impacts Earth and creates debris that soon coalesces into the Moon.

But there is something I did not find mentioned: How would the Moon get its circular orbit?

A debris field from any impact would at first follow a very elliptical orbit, with a perigee very close to the surface of the Earth and an apogee much farther away. In orbital mechanics, where you start is where you end up. In fact it doesn't even have to be where you start. Pick any point in the orbit and you will return there again. In other words, the original elliptical orbit doesn't change its shape.

...Unless there are pertubations or impacts that change your velocity. How exactly did that happen? Does the GIH just assume something else did that fortuitously?

I've also seen youtube video simulations like this. It's hard to tell if the camera angle is changing or if the orbital plane is unstable (how?), but you can clearly see the debris goes around and impacts the surface again and again. I.e., the orbit remains highly elliptical. I've never seen a simulation that shows the debris circularizing.

I also want to note that this question is highly relavent not just to the Moon, but many other things like orbital capture as well. I've never understood orbital capture theories as they all seem to leave out how the initial elliptical orbit later becomes circular, e.g., Triton.
I want to point out that the Moon's orbit isn't circular now.   A 0.055 mean eccentricity isn't that circular.  

But, onto your question.   I think you're making a bad assumption on the "must have started very elliptic.   Individual objects that are ejected from a planet need to follow their orbital path.   So any object flung from the earth would need likely either escape the Earth or fall back into it because you can't launch something, with a singular push, into a circular orbit.   All the individual bits of debris should have had highly eccentric orbits.

But the formation of the Moon was much more complicated than that.   First, it was enough matter to have it's own gravitational field, basically influencing itself, and the forming moon would be the combination of all those individual objects, so it wouldn't necessarily be eliptic at all.   The eccentricity of billions of objects could have (and likely did) cancel each other out to a large degree.  

Because the impact was off-center, it set the Earth spinning and most of the debris moved in the same direction around the Earth.   But the coalescing mass of material that didn't fall back to the Earth could have been in a reasonably circular orbit at formation.  

We should also define what number constitutes "highly" elliptic.   The Moon is thought to have started out 3-5 Earth Radii distant.   Any material that passed closer than 3 radii would have been inside the Roche limit and had a hard time coalescing into the Moon.    This article suggests that the debris would have had a hard time being ejected further than 5 Earth Radii, though I'm not sure how that conclusion was reached.  But you have a few things going on.  Debris colliding with other debris, and some (presumably quite a bit) of debris falling back to the Earth and, mentioned above, the debris having a gravitational effect on itself.   Ultimately momentum needs to be conserved and calculating the formation of the Moon requires a super-computer, but in general I think the initial orbit could have been relatively circular.

If the Moon formed close to the Earth, which most models suggest it did and since it couldn't pass inside 3 Earth Radii and stay solid, those numbers put a limit on how eccentric the initial orbit probably was.   If we use the 5/3 Earth Radii as an estimate, Ra=5, Rp=3, then the eccentricity is 0.25, and that's an estimate for the upper limit of the initial eccentricity after formation.  It may well have been quite a bit less.



But for the sake of argument, lets give the newly formed moon an eccentricity of .25 or, maybe a little higher with a more distant apogee, but we have to keep the perigee at 3 Earth radii or greater.   Your question still stands, how did the Moon's initial eccentricity become somewhat circular.

The answer is tidal circularization.   I can't find a good article on it, but for many 2 body systems, tides circularize orbits.   It's well known that the tides on Earth, caused by the Moon's gravity rotate ahead of the Moon and this creates a tug on the moon that pushes (pulls?) the Moon into a higher orbit, further away from the Earth.   This same secondary effect circularizes the Moon's orbit because the closer the Moon is too the Earth, the greater the push.  The math behind this gets very complicated and it's above my paygrade to put numbers behind it and the Wikipedia article I've linked is very lacking, so I invite anyone to provide more details if they can. But Tidal circularization is a commonly accepted theory and given the strength of the tides early on, it's likely any eccentricity in the Moon's orbit circularized relatively quickly, at least, astronomically speaking.   (Tens of millions of years is a long time to you and me, but not to a stable orbit).