When Maxwell's equations are solved, one of the solutions is electromagnetic waves that should move at a certain speed ($c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$). Now, one could argue that since Maxwell's equations hold for all observers regardless of their reference frame, they should all see these waves with speed $c$. So, the speed of these waves must be independent of your reference frame.
Moreover, let's say you could observe someone travelling at more than this speed of electromagnetic waves. Now, someone in this reference frame produced these waves. Also, there is a wall in front of them that is destroyed if the electromagnetic wave touches it (it is a powerful laser). And if the person in this reference frame hits the wall, he dies. Now from your perspective, he will die since he is travelling faster than the wave and will reach it first. However, the person himself would be sure he could survive if he fired the electromagnetic wave at the wall before he got to it. So, he would be sure he would not die.
This leads to a contradiction and so it must be impossible to observe a reference frame that travels faster than this speed of the electromagnetic waves.
I suspect there is probably a hole in this line of argument but I can't imagine what it is.
Best Answer
To answer your title question, definitely not. (Note: The title question has since been updated. The original was - "is it possible to deduce the constancy of the speed of light from Maxwell's equations?")
If you assume Maxwell's equations hold for all inertial frames as you are doing in your argument, then you are begging the question. You are making an assumption further to Maxwell's equations. That assumption cannot be deduced from Maxwell's equations themselves - you then need to ask how Maxwell's equations transform between different reference frames and that answer is outside the scope of Maxwell's theory.
In the 19th century, as hinted by dmckee's comment:
people simply assumed that Maxwell's equations held for an observer who was stationary with respect to the luminiferous aether, and that this aether would behave just as gas or the wood of a violin's sounding board for sound. Maxwell's equations would then change their form under the Galilean transformation. That was just "what waves did" to the mind of a physicist in 1862.
Physicists also assumed that Galileo's postulate, that only relative motion between observers were experimentally detectable, didn't hold for electromagnetism. Galileo, after all, knew nothing of electromagnetism.
And that is a perfectly plausible theory to postulate: from a logic standpoint, it is just as valid as Einstein's second postulate and the restoration of Galileo's relativity principle. These things cannot be settled by logic, but only by asking Nature for her take on these things. That is, they can only be settled by experiment - and we find that Maxwell's equations do indeed transform covariantly and that the transformation between relatively moving inertial frames is the Lorentz, not Galilee, transformation.