What is so special about a melodic half step that makes it "want" to go up (As in the leading tone to the tonic). Why isn't there as much resolution in a melodic half-step downwards?
This is certainly not universally accepted, but I say a half-step isn't really a fixed entity; rather there are a couple of different "half"-steps of different size that can serve different purposes.
That particular leading vii-I step, according to some performers – Pablo Casals was perhaps the most radical propagator of this idea – should be extremely narrow, i.e. the vii so high it can't possibly form a harmonically "steady" rendition of the dominant V chord (even if nobody plays that chord, the listener might infer it). This sort of already-too-high note then, according to his "gravitation" hypothesis, suggests an upwards force that naturally requires the note to be resolved to the nearby pure I tonic.
So basically it's not the vii note itself that requires upwards resolution, it's its dissonant harmonic context. There are a couple of ways this can be achieved, even if playing an equal-tempered vii note. The dominant can be a seventh-chord, whose tritone certainly creates some dissonant tension, between the IV and vii steps. Indeed that not only "pushes" the vii up resolving to I, it also pushes the IV down, resolving downwards either a half-step to the major tonic's iii, or a whole step to the minor's III. These down-resolutions certainly are important as well, but since the iii is a somewhat less satisfying final note than I (in particular if rendered in 12-edo, where this third is as much too-high compared to just intonation as the vii within the dominant), it will more usually be implemented in some middle voice rather than the main melody.
Well, the reductive answer is that there's nothing special about it and that the only thing that makes a leading tone "want" to resolve to the tonic is hundreds of years of musical convention, since this tendency exists in a Western scale but not necessarily in other scales.
The tuning of the Western scale has changed a lot over the course of those hundreds of years. Historically, for example, flats tended to be higher than sharps (that's a gross oversimplification but it'll do--from Leopold Mozart, for example: "according to their proper ratios, notes with flat signs are a comma higher than those in the same position with a sharp sign. For example D flat is higher than C#, A flat higher than G#, G flat than F# and so on"), but now, with "expressive intonation," sharps tend to be higher than flats (usually about a seventh of a tone). For a fuller discussion you could take a look at the book How Equal Temperament Ruined Harmony (discussed more fully here)--I don't agree with everything he says but it's a good introduction to the ideas involved.
There are theoretical reasons why this occurs that are not conditioned responses.
The overtone series if the fundamental expression of "music". That is, the overtone series is a natural occurrence due to the way the universe is designed. I won't get into the why's here but basically it is physics. I can explain more if need be.
The overtone series of the pitch C is C C G C E G Bb C D E F# G A B C. That of F is F F C F A C Eb F G A B C D E F. (each of these are not technically pitches but sinusoidal frequencies, I won't make a distinction here)
These come from nature and our ear is used to hearing these relationships produced by the overtone series is everything. Most of the higher harmonics are subtle but we have been conditioned by evolution to hear the relationships in the overtone series as "natural". (since the OT series is natural itself)
Now, in the OT series we have several observations:
The P5 down is a very fundamental relationship for motion between sounds. Why? Because C both has it's own overtone series and the next strongest OT series it is a part of is F, a P5 down. We also find the P5 in the OT series to the first new tone. The P4 is derived from this through the inverse relationship.
We find that the b7th occurs way more closely than the n7th.
The n7 occurs in the context of the major scale found in the higher overtones. The OT series can be thought of as a C7#11 + C major scale. In this case the n7 is dissonant with both the F and Bb in the OT series, the Bb being natural because F is related to it through it's own OT series.
Since the n7 is highly irregular in the OT series, that is, has little relationship to the other tones, when it is used in the as a scale, we our minds have to resolve the inconsistent use. We expect a Bb but we get a B, which is very close to C.
Our "ears" and mind have to/try to make sense of what we hear and use the build in OT as the "rules" to make sense of it. The B has to either move to Bb or C, and since our tonic is C, our mind wants the B to be a the tonic. The B sounds like an "out of tune" C more than it does anything else(not in the sense beats but of psychological dissonance).
While there are a few assumptions, they are very natural and probably how things have worked out. Just because other cultures do not fit this mold says nothing as they can be wrong just as they could be right. I won't get into this discussion here though as people tend to get emotionally involved...
Imagine a growing living organism growing from a single cell. The physical environment around it consists of "sounds". Many of these sounds produce a very specific "color", the OT series. As evolution goes, the organism slowly evolves to sense these colors and other colors in it's environment. Since most vibrating things produce sound approximately like the OT series, it is natural and correct to expect that the organism will develop senses to recognize the OT series and over millions of years the OT series will become fundamental to how it "hears" sound.
That is effectively what has happened, more or less. The OT series is built in to all living creatures just as gravity, temperature, etc. Hence fitting music "naturally" with the OT series creates natural music. Western harmony is more natural in this sense because many of the rules it uses are derived from the OT series. This is not about right and wrong, so don't get your panties in a wad ;)
Other cultures that are less "refined" harmonically(which is virtually all "non-western music") usually have stuck to more basic relationships found in the OT series and have not incorporated some of the more distant relationships... usually having advancemed in other areas such as rhythm or ornamentation, etc.
Ultimately, as human music evolves, it will evolve on many dimensions. Rhythm, for example, has it's own basis in reality analogous to how harmony and melody are based in the OT series. Western culture has generally stuck to the more basic rhythmic relationships. Generally this is due to "information theory" and "complexity theory". As harmonic relationships become more complex one sacrifices complexity in other areas else the whole is too complex to be understood in the first place. It seems different cultures balance the equation differently and get different results. This is a good thing IMO but we should not pretend that all are equally valid in the same respects, else we lose track of what they actually have achieved.
There's no special strength in a rising half-step. There IS special strength in a leading note resolving to a tonic, particularly when it's reinforced by the note a tritone interful below. Then you have the bare bones of a dominant 7th chord, and dominants tending towards tonics are the engine of all Common Practice music. It's not the only way to construct music, but it's a very powerful one!