For low photon energies, the behaviour is the same, although obviously the different mass of the proton and electron will affect the precise details.
At higher photon energies things change substantially, because the unlike an electron, a proton has structure and isn't a fundamental particle. In very broad terms, once the photon energy becomes comparable to the binding energy of the proton, things start to change, because a) the fact that you have three point charges (the discerete quarks) rather than a single point charge starts to matter, and b) you are dealing with bound charges, so consideration of excited states, energy levels etc. come in to play.
As an analogy consider scattering of photons off bound electrons in atoms. In the least Compton-like scenario, a photon is absorbed raising an electron to a higher energy state, and it then radiatively de-excites emitting multiple lower energy photons. This is nothing whatsoever like Compton scattering.
(Although I can't pretend to know enough about QCD to have any idea of what energy levels of quarks in a proton might look like, or whether there are in fact any).
Leaving aside the issue of whether an electron is a fundamental particle or not - it doesn't matter as long as the energy of the photon scattering off the electron is $\ll$ the binding energy of whatever constituent particles an electron might be made of, and at the highest energy interactions we've observed there is no indication of any electron structure.
There are additional complications for very high energy photons anyway, which also apply to Compton scattering, such as the probability for particle - anti-particle pair production which means that Compton scattering isn't the only thing going on once you get above the pair production threshold.
I'd suspect that that would actually come in to effect before any proton structure effects did, so as long as the photon is below the pair production threshold ($\approx 1$ MeV), protons will behave like a single point charge, much like an electron, aside from the mass.