I am planning to go to graduate school for mathematics in a few years. I want to know which math graduate school(s) have the most rigorous math courses. I live in the USA, so I would prefer an USA math graduate school. By rigorous, I mean that all the statements, except the axioms, are proven.
I think this question is misguided. It turns out that if you want to prove "everything from the ground up", then mathematics is actually exceptionally boring -- because you will never get very far. At least as a student, you have to be able to rely on statements others have proved and for which you may or may not understand the proofs to do interesting things, or at best you sometimes have to go through a proof in some sort of cursory way without dwelling on all of the details (and without wondering whether this step might rely on the continuum hypothesis and/or might only be valid based on some system of axioms but not another).
This isn't to say that mathematics should be all handwaving. But mathematics becomes a lot more interesting if you allow yourself to use tools you haven't developed yourself and might not actually be able to develop yourself. The point being that just because you don't understand doesn't make the theorem it proves any less valid: Someone else has already done the work for you, and you should feel free to use it.
This perspective is really not so different from saying "Yes, I could try to learn everything about cars and build my own. But my road trips will be far more interesting (and be able to cover far more ground) if I allow myself to buy a car from an established car company whose cars are known to last for a couple 100,000 miles." You don't have to be able to build a car yourself to drive one!
I doubt you will find such a place. In part because "proof" is somewhat in the eyes of the beholder. But any decent graduate school in the US or elsewhere will serve you well.
However, you will find some professors scattered around at those schools that require more from students than the average. There are still some folks around who studied with Robert Lee Moore at UT Austin, though they are mostly my age (i.e. retired).
The Moore Method of teaching is to forbid students from reading mathematics, but to develop it completely on their own. He didn't develop everything from the axioms, but his students did.
Some other younger people still carry on that tradition, though I suspect it is fading. It is a very hard path, though some people like it. But some of his own students have rejected it, also.
OOPS, Most, perhaps all, of Moore's direct students are now deceased. So look for their students, who might be carrying on the traditions. A place to look is in the Mathematics Genealogy Project
Somewhat as @Buffy says, your concern about careful, complete proofs in graduate school is probably misplaced. The top 100 or 200 graduate programs in math in the U.S. are staffed with good mathematicians who know how to really prove things, etc. :)
More to the point: there are many textbooks and on-line notes that fill in many, many details of many, many things, so there is no obstacle to you finding good proofs of nearly anything you want.
Especially nowadays, as opposed to pre-internet times, but already if a good textbook was available, lectures might not explicitly cover every detail, since it could be looked-up. Further, indeed, a lecture is a different medium/format/vehicle than written things... and one need not attempt to be a bad copy of the other. There is not much reason for a lecturer to copy onto a blackboard (or overheads, or ...) from their notes, to be copied by students... if/when an accessible type-set document exists. So, for me, I do not prove everything in class, leaving some to my typeset, on-line notes.
Especially, delicate arguments that are not very conceptual are probably better communicated in writing, with live discussion addressing just the delicacy, but not attempting the execution.
(For that matter, "in real life", mathematics is not actually lined up in impeccable logical order. First, because there is no unique such order... )
Anyway, I think you'll not find any real problem of the sort you fear. As in a comment, the aspects in which grad schools (in math, in the U.S.) differ are more in terms of the local culture, your peer group, etc.