Q:

Could a strong enough electric field tear a hydrogen atom apart?

A neutral hydrogen atom is composed of a proton and an electron.
The overall charge on the atom is zero but there are local charges within the atom as the negative and positive charge is not evenly distributed.
If a neutral hydrogen atom were placed in an electric field the electron and the proton would experience a force in opposite directions.
Could a strong enough electric field tear the electron and proton apart?

A:

Further to freecharly's nice answer, it's important to note that the use of the tunnel effect is not really necessary, and if you make the electric field strong enough, then the barrier will disappear, and the electron will leap out (in a process called over-the-barrier ionization) with extremely high probability. In the limit of extremely high electric fields, it will become impossible for the electron to stay near the proton, and the bound states of the system will disappear.
Now, that said, the electric field strengths $E$ required to get either of these processes in motion need to be such that the potential energy $eE\,a$ acquired over the atomic scale $a$ is of the order of the atomic ionization potential $I_p$, i.e.
$$
E\approx \frac{I_p}{ea} \approx\frac{10\:\mathrm{eV}}{e\cdot0.5\:Å } \approx 10^{11}\:\mathrm{V/m},
$$
are many orders of magnitude too big to produce using static fields, but they are available if you use light, instead; in that case, to achieve tunnel ionization you typically need intensities of the order of $10^{14} \:\rm W/cm^2$, while over-the-barrier ionization tends to need light closer to $10^{15}$ or even $10^{16} \:\rm W/cm^2$, depending on the atom in question. These intensities are by now quite routine in strong-field-physics laboratories around the world, and tunnel ionization is a crucial component of the physical processes that underlie technologies like high-harmonic generation and laser wakefield acceleration.
It's also important to remark that the fact that the electric field is oscillating does introduce some nontrivial alterations to the dynamics (see e.g. the Wikipedia article on tunnel ionization for some more details on that), but these go away if the laser's period is long enough (or if the laser is strong enough that the ionization happens too quickly for the electron to notice that the field is oscillatory). Experimentally, most strong-field lasers operate at around 800nm, which is already very slow (so, in particular, its photon energy is tiny, and the picture of ionization as a multi-photon process becomes less useful), and there are multiple high-intensity laser systems getting built at 2μm and longer wavelengths, where the field can to a rather good approximation be seen as static as far as the ionization step is concerned.

A:

A hydrogen atom can be "torn apart" by a strong enough electric field. This means that the electron of the hydrogen atom will leave the proton. This phenomenon is called field ionization which is caused by a quantum mechanical tunneling process where no energy needs to be supplied to the electron. Without applied electric field, the electron sits in its lowest energy state in the Coulomb potential well of the positive nucleus. If a homogeneous electric field is superimposed on the $1/r^2$ Coulomb field, a potential barrier forms. When at sufficiently high applied field this barrier becomes very thin, the electron can tunnel through this barrier (tunnel effect) and leave the nucleus for good.  

