In normal usage, an axiom is something that is self-evident or tautologically true. For instance, if we think about the algebraic reflexivity axiom (a=b ⬄ b=a) or Euclid's postulate that any line segment can be extended indefinitely, it's clear that these are not things we actually prove, but rather things that we point at and say 'Duh!'. We accept them mostly because rejecting them makes mathematics impossibly counter-intuitive.
The social world is at least an order of magnitude more complex than the mathematical world. The principles that lie behind a constitution are often presented in the form of axioms (particularly in the Jeffersonian "We hold these truths to be self-evident" language), but that is a claim to moral certainty, not a tautological truth. If one accepts the moral framework that underlies the constitutional structure, then the moral claim is axiomatic. But we humans are notoriously fickle about moral frameworks — some of us more so than others — so the claim to self-evidentiality is aspirational at best.