After some research, it seems apparent to me that the EM spectrum is continuous, but this would contradict a physics fundamental that energy is discrete. Is there a conflict here?
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No, there isn't a contradiction. From the very inception of the idea of quantization of energy (in Planck's explanation of the continuous blackbody spectrum) the two ideas have been compatible.
The "physics fundamental" that "energy is discrete" that you refer to, when applied to the electromagnetic field, says that each mode of electromagnetic radiation at frequency $\nu$ can only hold energy in discrete lumps of size $h\nu$. However, there is a continuous infinity of possible modes spanning all possible frequencies $\nu\in(0,\infty)$.
There are additional aspects of energy quantization when matter is involved (most commonly the photoelectric effect and the discrete nature of atomic emission and absorption lines) but these refer to the nature of the energy transitions that matter itself can perform and interact with, and they say nothing about the nature of the frequency and energetic spectrum that the EM field itself can accommodate.
Emilio Pisanty
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