Diagonalization proceeds from a list of real numbers to another real number (D) that's not on that list (because D's nth digit differs from that of the nth number on the list).
But this argument only works if D is a real number and this does not seem obvious to me!
Especially given that in Cantor's time there was no developed theory of computability, I could imagine someone of that era saying "ah, Cantor, you are pulling a trick here! D seems like a real number at first glance but when you look closely it's infinitely difficult to calculate or really talk about with precision in any way. In this regard it's unlike every real number I've ever encountered and you must make an additional argument to make me consider it a genuine real."
My question is this: was Cantor's diagonalization argument—at the time he made it—rigorous? Either in the sense that it proceeds from axioms or in the sense that it contains a convincing answer the above skeptic? Does D's real number-ness require a justification and did Cantor offer a sufficient one?
Of course I am not questioning whether the reals are actually uncountable. I'm interested in extent to which Cantor actually proved this versus giving an intuitive argument in favor of believing it.
You seem to be assuming a very peculiar set of axioms - e.g. that "only computable things exist." This isn't what mathematics uses in general, but even beyond that it doesn't get in the way of Cantor: Cantor's argument shows, for example, that:
This is roughly because we can compute the $n$th bit of the antidiagonal real by looking at the $n$th bit of the $n$th real on the list. (I'm hiding some details about numbers with two decimal expansions here, but they're easily overcome.) So even if you only believe in computable objects - which, again, isn't what mathematicians generally do - you're stuck with Cantor.
Put another way, if you want to restrict attention to computable objects, then you're stuck with only computable lists as well. And at this point Cantor's theorem is just the usual computability-theoretic fact that there is no computable enumeration of all the computable reals.
What we're seeing here is that Cantor's argument applies in pretty much any coherent "mathematical world" - restricting attention to "concrete objects" doesn't break it since the "antidiagonal real" is as concrete as the list it "responds" to. One way to make this observation precise is via category theory, where we can observe that Cantor's theorem holds in an arbitrary topos, and this has the benefit of also subsuming a variety of other diagonalization arguments (e.g. the uncomputability of the halting problem and Godel's incompleteness theorem).
Second, let's address the historical aspect. Cantor's argument of course relies on a rigorous definition of "real number," and indeed a choice of ambient system of axioms. But this is true for every theorem - do you extend the same kind of skepticism to, say, the extreme value theorem? Note that the proof of the EVT is much, much harder than Cantor's arguments, and in fact isn't computably true!
Basically, the issues with formalizing the reals and mathematics in general carry no more weight here than they do with other, apparently-less-objectoinable, theorems.