As a materials engineer the concept of a crystalline desert full of glass is baffling, as the two terms, crystal and glass, mean essentially opposite things. I'd like to start by making sure we are using the same terminology. I apologize in advance if I come across as lecturing! It is not my intent to bring bad feelings.
A crystal is most commonly defined as a periodic lattice with a repeating unit cell. A crystalline material would thus have its atoms arranged periodically on a regular grid in space and thus exhibit long-range order. If you were to move around the crystalline material, you would see the same atoms whenever you move one unit of distance along one of the axes of the lattice. Crystals are highly organized, and in thermodynamic equilibrium are the preferred state of all inorganic matter when in the solid phase.
An amorphous material is a condensed material that does not exhibit long-range order, and both liquids and solids can be amorphous. When a solid material is amorphous, it is commonly called a glassy material, glassy solid, or simply a glass.
Well, why does this matter?
Cooling rates! Allow me to explain:
I did mention that in thermodynamic equilibrium crystals are preferred for inorganic matter in the solid phase. However in practice kinetics can make achieving thermodynamic equilibrium occur over geologic timescales or longer. In the case of pure silica (SiO2) at room temperature, the kinetics of a glass to crystal transformation is very slow, and the timescale is on the order of hundreds of millions of years. It is difficult to measure the rate directly at room temperature because in a lab setting nothing measurable happens within a human lifetime.
The reason silica forms glass is that at an atomic scale, each Si4+ ion shares one electron with each of four O2- ions, forming a tetrahedral ionized molecule. Each O2- ion bonds with two Si4+, linking two silica tetrahedra at a common vertex. Because the oxygen ion bonds are relatively flexible, the tetrahedra have only a vaguely preferred orientation with respect to one another; very little energy is required to knock them out of alignment during cooling. Thus, as a body of silica cools from the liquid phase, it can take a long time for regular, periodic crystals to form and settle in, and instead the vibrating atoms may slow down in whatever arrangement the tetrahedra happen to be in, which may or may not be a periodic lattice.
Therefore, if the cooling rate is slower than the crystal forming rate, a crystalline solid forms. On the other hand, if the cooling rate is faster than the crystal forming rate, an amorphous solid forms: glass.
Now to address your questions:
What kind of temperatures would have to be achieved to do this to a
Assuming the entirety of your glass precursor material is pure silica (SiO2), the melting point is approximately 1700 C. Other components in solution with silica such as alumina, magnesia, and iron oxides will generally serve to reduce the melting point, however, alumina and iron oxides will make it much easier to crystallize the material as it cools. Beware, as you would want to maintain a temperature above 1700 C long enough to heat everything beneath the surface above the melting point as well, or you will only get a superficial layer of liquid.
In practice, artificial glass forming techniques take advantage of what are called network modifiers, which are metals with valences of 1 or 2, that disrupt the tetrahedral structure even further than it is in pure silica. This (1) reduces the melting point, requiring less energy to form the glass, (2) baffles the ability of the glass to crystallize, meaning even lower cooling rates are required to form glass instead of crystal, (3) reduce the glass viscosity at every temperature, making it easier to shape. So if you have a lot of Group Ia and IIa elements in your precursor material, e.g. lithium, sodium, calcium, magnesium, etc., then more glass will probably form from the same event.
One thing to worry about is that heating 250K sq. mi. to any significant depth involves a tremendous amount of heat, and the heat in that much volume will take a very long time to dissipate into the rest of the planet, its atmosphere, or out into space. We are talking somewhere in the range of thousands to millions of years depending on depth. As a point of reference, a 50,000 pound steel casting can take a day to completely solidify, and possibly a week or more to cool to shipping temperatures. We are talking about possibly trillions of tons of material heated to the same temperatures. In that time, it is possible for lots of defects to show up in the glass for a variety of reasons, most notably ejecta returning to the surface.
There is also the problem of crack formation from volume changes on cooling. Cooling materials decrease in volume as they lose heat, but the glass crater surface also wants to stay put due to friction from its own weight on whatever material it is sitting on. There is thus a competition between the force of cooling-induced shrinkage, and friction. Something has to give to balance forces, and so the material will fracture, and many cracks will form in a single-piece glass crater. Unfortunately, if allowed to cool under its own devices, in other words an excruciatingly slow cooling rate, most of these cracks are going to be small and spaced on the order of inches apart, effectively breaking up your glass surface into little chunks. However, if you can magically cool it instantly throughout its entire volume, it will form cracks only spaced very widely apart. It will also form a glass to a greater depth into the ground. With magic, you could potentially have what is effectively a crater filled by an ocean of glass.
My recommendation is to have a near-pure silica-sand desert, and a very large, very high-temperature magical effect (above 1700 C). Silica is incredibly common on Earth, so it is plausible if your planet is Earth-like. Very large meteor strikes would satisfy the magical heating effect nicely, as would a firestorm, or a radiant "explosion" of some sort. If you want large contiguous pieces of glass instead of a crunchy thin bed of glass fragments, try following the high-temperature magical effect with a rapid cooling magical effect. Faster cooling will give bigger, deeper chunks of glass.
Would the event have to take place in an area that was a desert in the
first place? Is sand required?
Sand is not required, but silica is virtually required for what one might call "practical" methods. With magic, other materials are possible.
So you could possibly get away with silica-based rocks, from which silica-sand would be made. Other oxide materials may form glasses, but are much less common on earth than silica. If your world is Earth-like, you may want to just stick with silica.
In principle, any inorganic solid can form a glass if you cool it fast enough. Unfortunately for us, the materials must be either cooled extremely rapidly (106 Kelvin per second or faster for pure metals), or baffled from forming crystals as with silica. It is possible to get metals to form glasses, but the entire volume of pure metal must be cooled more rapidly than is possible by any method known, with the exception of samples that have ~10 nanometer thicknesses or less.
Specialized, artificial, and often very expensive alloys have been developed which require cooling rates ranging from 105 K/s down to 1 K/s. Current iPhones have a small part (the sim-card ejector) which uses an iron-based alloy called liquidmetal that I would guess requires approximately 10 K/s to form a glass. It is also both incredibly strong and incredibly tough compared with crystalline iron alloys. However, as noted on their (site), production sizes are limited by the cooling rates achievable.
With appropriate magic, and a lot of processed iron metal, you could have a basin full of amorphous iron! What I wouldn't give to be able to make that happen on Earth, admittedly at smaller scales and in a controlled, repeatable and inexpensive fashion.
What would be the global impact of such temperatures in a localized area?
As noted above, all that heat has to go somewhere. A meteor creating a crater the size of Texas would cause global devastation and mass extinctions. The planet's climate would be altered for tens of thousands of years or longer. It is implausible survivors would ever see the glass basin before it is broken up and buried by geological action. It is arguably implausible for there to be human survivors with any notion of civilization without some serious magical intervention anyway.
On the other hand, if it is some sort of radiant heating effect, it would be more locally destructive, but would still create massive weather effects including likely global storms, evaporation of nearby bodies of water causing more overcast conditions and rain elsewhere in the region or world, earthquakes from the sudden change of the shape of the ground as well as the huge expansion of matter from heating. Global storms could create problems on the ocean as well, including larger than normal waves. There would be noticeable effects worldwide even with a radiant heating effect, though to a much less severe degree than with a meteor.
Alternatively, you could, as I've mentioned, use a rapid cooling magical effect to remove a lot of the heat instantly, though I suppose this may not fit into your world. If you choose this option, however, that would be sufficient to explain away a lot of the problems associated with a sudden influx of heat and energy into a large chunk of your world. There would plausibly be mild to no long term climate or weather effects in such a scenario. It's possible people thousands of miles away would hardly notice. However, it should be obvious that no matter how this happens, everyone and everything in your Texas will be vaporized or melted into an unrecognizable and meaningless state.
The way I would do it is using molecular-scale magic effects. I recommend a widespread molecular heating effect sufficient to heat a large grassy region down through much of its bedrock. This would then be followed up immediately by the exact same effect but with cooling instead of heating, and much faster (read: instantaneous). This is effectively the only way to make this result in large contiguous chunks of glass while staying consistent with Earth-like physics and material science.
On a side note, this is starting to sound like a scene in a novel set in the Dragonlance campaign setting circa 2nd edition that I read 15 years or so ago. Unfortunately I don't remember the title of the novel, or any of the character names, so I can't really help you identify it, unfortunately. In the scene, a character was walking through a(n in)famous region where a powerful magic-user had used very large-radius, powerful, elemental magic to create a firestorm and an icestorm (possibly among other effects), to stop an opposing army, with the icestorm last. The freezing effect ended up being permanent, and the magic-user and what was left of the army were frozen in place forever. So, a bit similar to what you're talking about, but with ice instead of silica glass.
Edit: I forgot XKCD What-if? has this article, the latter half of which describes the ancient Chicxulub impact (Wikipedia). The area of the crater is considerably smaller than Texas, at about 5%. Munroe describes ejecta reaching space in a similar explosion.