Very briefly (and disappointingly): no, or at least not in the vast majority of cases.
Identification of geographic origin for most minerals (and gems) is made possible by the study of inclusions and impurities in the main stone and it. Since in most locations the mix of minerals and impurities is unique, it makes identification or at least a very good guess possible. Diamonds are one of very few minerals for which this does not work.
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For example Burma rubies have a unique "signature" consisting of inclusions of rutile needles, calcite, spinel, pyrite and sphene. Thai rubies generally reveal through spectroscopy the presence of iron - sometimes giving a brown/orange tint - but never present the inclusions typical of Burmese material, having instead minute crystals of apatite or very characteristic laminar fissures with liquid inclusions. Sri-Lankan rubies have less uniform colouration (and sometimes a bluish/purplish tint) and have commonly two-phase inclusions (bubbles containing gas and liquid), together with mica and hematite crystals and iron titanate "lumps", as well as rutile but in a different form from Burmese material, since it is larger, thicker crystals rather than thin needles.
Unfortunately, diamonds (and a few other minerals, like olivine/peridot) are not created trough the same process as rubies, in the upper layers of the Earth's crust. They are born much deeper in the Earth, at several tens of kilometres into the mantle (the semi-liquid layer below the crust). This creates two problems: firstly, because the mantle is far hotter than the crust (it's basically molten rock), most minerals melt and no longer can be identified. Secondly, because of the liquid nature of the mantle, even high concentrations of impurities tend to diffuse relatively rapidly before the material is ejected to the surface through volcanoes.
This creates a much greater uniformity in the characteristics of diamonds, and even the relatively few indicators that are left are not at all proof of origin, since they seem to depend on accidents specific to an individual stone. For example, there is a tendency for South-African diamonds to contain nitrogen and thus tend to turn to a yellow colour (so-called type I diamonds), while old Indian material is poorer in nitrogen, giving rise to the name "Golconda diamonds" for stones that contain no nitrogen (type II diamonds). However, the Cullinan and several other large stones of South African origin are Type II, and there is no doubt that several "Indian" stones from the 18th century were originally coming from Brazil: they were transported to India and sold there because the merchants realised that Indian diamonds would trade at a higher price!
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In practice, this means that unless a diamond is tracked from the mining point forward, it is not possible to know where it came from by means of chemical/physical examination afterwards. Some companies - for example the Argyle mine in Australia and some Canadian mines - make a point of doing this (and managing to charge a premium for their stones), and important stones are often (but not always) reported and tracked individually.
In recent years, concerns about the use of diamonds to fund war activities (blood diamonds) have led to the creation of tracking processes ("Kimberly process") to ensure that diamonds entering the global market come from "clean" sources. However, 1) the processes have proven to be porous and not able to guarantee provenance, letting quite a few "bloodied" stones through, 2) the processes are not aimed at distinguishing geographical origin, only whether a stone is clean or not, and 3) they are totally non transparent to the consumer (or retailer, or wholesaler up to quite a high level in the supply chain), who only gets told "this diamond is certified through the Kimberly process" (or not), but is given no further information.
So, if origin of your diamond is important to you, you can either buy a diamond that is tracked from the mine onwards or go and dig out your own