No, Hawking radiation is based on well-known physics (virtual pair particle-anti-particle creation) which arises near an horizon (which you can treat classically: the so-called quantum foam play no role).
Simplified, hand-waving explanation: virtual pair creation can occur in any quantum field theory. We have quantum field theory describing three of the fundamental forces (strong, weak, electromagnetism ... ok, two forces as the last two are unified as electroweak). Pair creation can occur at a (length) scale that is much larger than the Planck scale. (the energies involved are much lower than that of the Planck scale). Treating a black hole classically, one particle from a virtual pair can cross the horizon and the other escape at infinity - thus you get Hawking radiation. At those scales, the event horizon would appear to be smooth - like a pure 2-d surface.
However, for gravity, we do not have an experimentally verified quantum theory. It is thought that such a theory would contain the equivalent of virtual pair creation, namely fluctuations in the space-time. So, at the Planck scale, one would expect that space-time would not be smooth and thta you would have spontaneous appearance of quantum fluctuations that would give your space-time a foamy structure. However, these structures would be incredibly tiny and, for "macroscopic objects like a proton (!)", the space-time would still look smooth; fluctuations of the even horizon of a black hole would occur at much smaller distance than those involved in virtual pair creation.
