I wouldn't say it's the strongest objection to wave theory but it does make a strong point. (I'd say Planck's blackbody curve does that much better.)
Specifically, if wave theory was a "good" solution then any frequency spectrum of the incoming wave should induce a reaction. So quanta would not do the job properly.
I want to go on record saying this, though: Wave Mechanics will work on this since there is essentially no difference at all between the results of wave theory and particle theory. (I admit I don't know how to successfully use wave theory here.) It's just that it is much easier to apply particles to explain this one.
Classical wave theory circa early 1900's predicted that the energy absorbed by a light wave was proportional to the intensity of the wave with no dependence on the frequency. The frequency threshold/work function in the photoelectric effect shows that this is not the case.
There is quite a lot of evidence that was found at around those times that demonstrated issues with the "classical" theories, such as the black-body radiation curve for emission, the ultraviolet catastrophe, properties of low-temperature gases (e.g. specific heat capacity), Young's double-slit, spectral line emission and absorption, semiconductors, superconductors, superfluids, etc.
Since those times, physicists have been able to explain the idea of a photon in wave theory using "wave-packets". Although this is interesting, quantum mechanics (QM) still reigns supreme because of the other observables.
I have seen several examples in Physics where two seemingly different mathematical descriptions have been proven to be exactly equivalent,
and many more where they have been shown to be very close approximations to each other.
However, one description may have the edge over the other for a number of reasons;
Perhaps one is easier to understand than the other,
Perhaps one mathematical description provides clearer hints regarding possible progression to the "next level" of physics,
Perhaps one more closely matches what people are familiar with than the other;
Perhaps (sort of Occam's razor style) one requires fewer fiddle factors than the other.
So as TopsQuark & Benit indicate, wave theory can be made to fit, but it becomes a bit awkward...