Maxwell’s Equations contain some other Laws as special cases. Does it possibly include the Ideal Gas Law as well? – the idea being that material substances can be modeled by appropriate subsets of the electromagnetic spectrum.
Maxwell's equations in their original form are scale-invariant and don't admit any solutions corresponding to massive particles; the only particle-like objects resulting from Maxwell's equations (in the vacuum, i.e. in the absence of pre-existing materials) are electromagnetic waves (reinterpreted as photons in the quantum theory) which move by the speed of light, and they're not what we call the ideal gas (which is reserved for a non-relativistic gas).
The ideal gas is composed of slowly moving (approximately) non-interacting particles such as atoms or molecules of gases and their properties have nothing to do with Maxwell's equations of electrodynamics. However, many of the properties of the ideal gas happened to be understood by the same James Clerk Maxwell. That's, for example, the reason why we talk about the Maxwell-Boltzmann distribution which governs the likelihood of different velocities of atoms or molecules in the ideal gas.
But this discovery belongs to statistical physics and has no direct relationship with the equations that govern electromagnetism – except for the sociological ones (both were co-discovered by Maxwell and both belong to pillars of classical physics).