Does the equivalence between inertial and gravitational mass imply anything about the Higgs mechanism? For example: the role it might play in a theory of quantum gravity (ie causing space-time curvature)?
I realize that inertial mass can result from binding energy alone. Has the equivalence principle been tested on elementary particles (like the electron) whose mass would be entirely due to the Higgs coupling?
It seems like an obvious question, but I have never heard it discussed before.
 A: Dear user,
the equivalence between the inertial mass and gravitational mass tells us the following thing about the Higgs mechanism:
Any inertial mass produced or modified by the Higgs mechanism also has to produce or modify a source of gravity of the same magnitude. And vice versa, a gravitational mass produced by the Higgs mechanism also has to produce an inertial mass of the same magnitude.
The reason why others - and myself - and confused about your question is that the equivalence principle tells us the very same thing not only about the Higgs mechanism but also about confinement, electrostatic attraction, spinning gyroscopes, or any other process, object, or mechanism that is taking place, has been taking place, or will take place. The equivalence principle is a totally universal law that holds for all objects and all processes in this Universe - and beyond.
It is not true that the Higgs mechanism has a more special relationship to the equivalence principle than any other mechanism in non-gravitational physics.
Concerning your more specific question about the experimental tests - of course that it has been tested. The available experimental tests of the equivalence principle show that all materials have the same ratio of inertial and gravitational mass up to the precision of $10^{-15}$ or so. Different materials have a different percentage of their mass coming from the electrons - the more neutrons a material has, the smaller the fraction is. So the mass stored in the electrons may go from 0.02% to 0.05%. While this is much smaller than 100%, it's surely enough to exclude the conjecture that the electron mass - as produced by the Higgs mechanism - doesn't obey the equivalence principle. 
The percentages above are just 3.5 orders of magnitude below 100%. So you still have 12 orders of magnitude left that prove that the mass produced by the electrons' interactions with the Higgs is exhibited both as inertial mass and gravitational mass - with the same ratio (one - in normal units) - as all other objects have. So once again, yes, all methods to obtain energy/mass i.e. $E=mc^2$ obey the equivalence principle and this fact has been tested with an amazing accuracy. The equivalence principle is true for masses produced by the Higgs mechanism, confinement, or anything else.
This fact is a problem for some "cheap" methods to solve the cosmological constant problem. The problem with the cosmological constant is that even things such as virtual objects in atomic physics, QCD, or anywhere create sources of vacuum energy density. It is not possible to throw them away because such a procedure would ultimately contradict the equivalence principle. So the mystery is why the cosmological constant is so tiny even though we may enumerate lots of possible sources that are much bigger and that appear at any conceivable scale.
Cheers
LM
A: The equivalence between inertial and gravitational masses (the weak equivalence principle) strictly holds only in metric theories of gravitation, such as Einstein's General Relativity. It may be violated e.g. in scalar-tensor theories of gravity (generalized Brans-Dicke theories) provided the scalar component non-universally couples to matter. The Higgs boson actually can play the role of such a scalar component mixed with the scalar graviton, since there is no reason to forbid a coupling of the Higgs field to the gravitational scalar curvature. Therefore, in principle, the weak equivalence principle is broken. However, with the reasonable Higgs boson - curvature coupling, this violation is practically unobservable, because of smallness of Higgs-Yukawa couplings to the light generation matter and short-range nature of the Higgs exchange force. 
