As Oddthinking said, this is a well-studied theory and the current consensus of the astronomical community regarding how the Moon formed. We have various bits of experimental evidence to support the Giant Impact Hypothesis:
- Oxygen ratios (16O, 17O and 18O) are very similar in samples from Earth and the Moon, but very different from other terrestrial bodies in the Solar System (Wiechert et al. 2001)1
- A thick layer of anorthosite has been found underneath much of the Moon, indicative of some large-scale melting event termed the lunar magma ocean (LMO), which can best be explained by the Giant Impact Hypothesis (Yamamoto et al. 2012)
- Recent lunar crust maps show that aluminum abundances on both Earth and the Moon do match; previous estimates had shown a large discrepancy (Wieczorek et al.)
- The Moon has higher ratios of heavy zinc isotopes to light zinc isotopes than Earth, confirming models that there must have been a period where zinc was evaporated, as expected from a giant impact (Paniello et al. 2012)
- The Moon formed about 30 million years later than most other protoplanetary objects its size, suggesting a non-traditional formation method (Stevenson & Halliday 2014)
- Furthermore, the Moon is receding from Earth, indicating it must have formed close to the planet.
- Clouds from similar collisions in other planetary systems seem to have been observed (Rhee et al. 2008)
There are a few more, including some other specific abundances of volatiles, but I think these are reasonably representative of what we have to support the idea. I'll quote Stevenson & Halliday by saying that the Giant Impact Hypothesis is the model that does the "least worst" job of explaining this all.
Then we have the simulations, which seem to do a decent job of reproducing the system's final state, including masses, angular momentum, etc. There are some disputes over the precise initial parameters (I think there's been some fresh dispute recently about the impact angle in particular, though I could be misremembering), but the results are still good. Some good theoretical evidence for the Giant Impact hypothesis includes
- Benz et al. 1986 (old and not state-of-the-art, but some of the earliest detailed simulations)
- Canup 2004 (one of a long series of simulations, and one that provided firm evidence for the current ideas of the impactor kinematics; Canup has done a lot of work on the Giant Impact Hypothesis)
Another 2004 paper by Canup (same guy) includes a nice overview of the Giant Impact Hypothesis in general, and provides context for much of the work done on it. It turns out that both of the above papers are cited in it.
1 Abundance ratios are slightly tricky because you might expect them to be different if the impactor came from another part of the Solar System. Some simulations indicate most of the material of the Moon came from the impactor, in which case similar oxygen ratios would mean it couldn't have come from elsewhere in the Solar System, which might be slightly problematic. It's open to interpretation. There are other solutions, too.