The first thing to say is that ordinary matter is actually not stable. Suppose a baseball-sized rock finds itself in the vacuum of outer space in the very distant future, isolated by the universe's accelerating expansion within its own cosmological horizon. Even within the standard model of particle physics, the rock will eventually decay by quantum-mechanical tunneling into more stable forms of matter. Over extremely long time scales, the result is believed to be that it will become a microscopic black hole, which then evaporates into other particles (mostly photons). (You will hear people say that this is the ultimate fate of all matter in the universe, which isn't actually right.) This kind of thing is discussed in Adams and Laughlin.
You asked about the stability of the hydrogen atom in various theories. There are some reasons to believe that the proton is unstable (google "proton decay"), in which case the hydrogen atom isn't actually stable. However, it is stable within specific models. Others have pointed out the Lieb paper, which in section I makes a specific technical argument about one type of stability for individual atoms according to one model. The model is the Schrodinger equation with a pointlike proton.
First off, there are really two things that are required in order to show that hydrogen is stable in this model, and Lieb only focuses on one of them, which is stability against a collapse of the electron's wavefunction so that it becomes bounded within an arbitrarily small distance from the proton.
The other type of stability that has to be demonstrated is stability against the electron's escape. Stability against escape is nontrivial. For example, the interaction between two neutrons is essentially purely attractive, and yet the two-neutron system is believed to be unbound. This is because the range of the force is so short (about $10^{-15}$ m). If the neutrons were to be confined within that distance of one another, they would have to have high kinetic energy, so they would fly apart. The reason hydrogen is bound is that the electrical force is long-range.
For hydrogen's stability against collapse, Lieb's argument is more complicated than it needs to be, because he unrealistically assumes a pointlike proton. Since protons are not really pointlike, compressing the electron to an arbitrarily small space $\epsilon$ near the center of the proton gives an electric field whose energy diverges to infinity like $1/\epsilon$. (If the proton were pointlike, then the external field would go to zero in this limit, so this argument would fail.)
Your question about quantum field theory is an interesting one. I think the nicest way to approach this is to look at the dimensionless and dimensionful quantities that you can form out of the relevant parameters. Most of the interesting physics can be understood in terms of two of these. There is the fine structure constant, $\alpha=ke^2/\hbar c\approx 1/137$, and the Bohr radius, $a_o=\hbar/mc\alpha$, where $m$ is the mass of the electron. In hydrogen, the typical velocity of the electron is $\alpha c$, and since this is small compared to c, you don't really need quantum field theory for hydrogen. The Schrodinger equation, which is nonrelativistic, is an excellent approximation. However, if you make a hydrogenlike atom consisting of a nucleus with atomic number $Z$ plus a single electron, the velocity in units of $c$ is on the order of $Z\alpha$. For large $Z$, this shows that you need relativity, and quantum field theory.
The Bohr radius is the only quantity you can form here with units of length. That suggests, without the need for explicit solution of the Schrodinger equation, that not only does hydrogen not collapse to an arbitrarily small size (as shown by Lieb's argument), but we expect it to reach a certain size which is basically the Bohr radius times some factor of order unity.
Adams and Laughlin, http://arxiv.org/abs/astro-ph/9701131
Lieb, Rev Mod Phys 48 (1976) 553, http://www.pas.rochester.edu/~rajeev/phy246/lieb.pdf
