This is a copy of a question I just asked at Physics Stack Exchange. From reading the text on the related questions, it seems that Maxwell equated light with the carrier of electromagnetic force just by the fact the two had the same propagation speed. That doesn't seem very rigorous; the two phenomena could have had the same speed by coincidence. Now, later scientists and engineers did find better proof connecting light and electromagnetism, but was Maxwell's assertion just a lucky guess? (AFAIK, light was the only known thing that had the propagation speed, but we have found other particles since then that share that speed.)
Maxwell had at least three arguments in favor of the conjecture on electromagnetic nature of light. The first one was philosophical (Chap. XX, section 781). He could not imagine waves propagating in empty space, so for the electromagnetic wave he had to assume the existence of some medium that fills the space. Then he writes:
To fill the space with a new medium whenever any new phenomenon is t be explained is by no means philosophical, but if the study of two different branches of science has independently suggested the idea of a medium and if the properties which must be attributed to the medium in order to account for electromagnetic phenomena are of the same kind as those which we attribute to the luminiferous medium in order to account for the phenomena of light, the evidence for the physical existence of the medium will be considerably strengthened
Then he derives the speed of propagation of electromagnetic waves, not only in vacuum but in different substances, so in fact he computes the index of refraction, in terms of electromagnetic constants, and gets a fair numeric coincidence (at least 2 digits in all cases).
And finally he discusses Faraday's experiments demonstrating the action of magnetism on light. Faraday was able to alter the plane of polarization of light by electro-magnet.
All these arguments discussed in Chapter 20 of his treatise make a very convincing case for electromagnetic nature of light.
I think it was speculated even before Maxwell that light might be an electromagnetic wave, or that there is a connection between electromagnetism and light.
It is true that Maxwell did not prove it, but - as you say - his equations for the first time rigorously predicted electromagnetic waves (and also all other electromagnetic phenomena). Perhaps the stronger indication is that it also predicted the speed of light correctly with only two constants (epsilon and mu) measured in static benchtop experiments. I would say this is - although not a proof - a strong indication and perhaps the people of that time also felt this way. And back than I think the speed of light was the only speed known to be this high.
To my knowledge electromagnetic waves were rigorously proven by direct experiment by Heinrich Hertz, which was after Maxwell write down his equations.