While I do not necessarily disagree with most of the points addressed in the other answers here, I do not think they actually answer the OP's question, which was clearly stated in their comment to @Manishearth's answer:
Hi. I don't mind taking the gravity equation as an axiom. What I want to ask (perhaps different from what the OP would like to ask) is: Suppose we assume certain axioms to describe the universe and use them to derive a statement P. Is the derivation of P from the axioms rigorous ? This is like the question: Is physics axiomatic ?
The answer to this question requires a nuance that I think is not being apprecaited in the other posted answers. Namely, what do we actually mean by "axioms" and "is the derivation of P from the axioms 'rigorous'$\,$"? These notions THEMSELVES were being developed in their modern form in the 20th century, with the rise of Hilbert's (among others) axiomatic framework, and exploded until today (Goedel's theorems, Zermelo-Frankel set theory, modern topology and graph theory, matroid theory, etc....). At the same time in the 20th century, physics was utilizing these new sophisticated mathematical machineries: for example, various approaches to quantum theory emerged, which Dirac showed to be equivalent algebraically, and the theory of general relativity used tensor theory, differential geometry, and Minkowski's spacetime formulation.
This historical context is important, I think, because it gives better meaning to the comparison between mathematics and physics. The two work together to advance each other - pure mathematicians probably won't agree, but the history of progress in math proves otherwise.
Therefore, I would say that YES theoretical physics - the part of "physics" where people build conceptual models using math - is axiomatic sometimes, not always, but it has been like this since Newton (there are possibly others that preceded him, but thats a question for History of S&M SE).
Newton's Principia is axiomatic and mathematically rigorous. He begins from well defined axioms (e.g., law of universal gravitation and laws of motion) and derives theorems and lemmas using calculus. For example, he showed that Kepler's heuristic rules of the solar bodies (which were built on Copernicus') are consequences of his laws. Here, Newton's laws act as the "axioms" of the theory.
But sometimes "axioms" in physical models are not called laws, and instead are principles, or even postulates. For example, Einstein postulated the "axioms" of special relativity, and uncovered a very enlightening model of reality. He then extended the special theory to the general theory with the equivalence principle, which is the conceptual core of general relativistic physics.
The laws of quantum mechanics are "axioms" in the exact sense that they are assumed to be true and then one deduces logical conclusions from them (i.e. theorems, etc...). As you say, if P is the proposition that "an electron has spin state $|\psi\rangle$" then you can use the laws of quantum mechanics to derive all kinds of other logical implications (i.e. theorems).
As @anna v stated, the axiomatic approach found use in physics mainly due to its ability to predict new phenomena, as well as generalizing already known information. Newton's laws were used by many in the centuries following him to predict the existence of many moons and planets. This predictive utility is easily seen in modern theories such as relativity and quantum theory: e.g., theory of general relativity's predictions of gravitational radiation, gravitational lensing, light bending, black holes, etc...., and Dirac's prediction of antimatter, the prediction of other fundamental particles, etc....
I think it's worth responding to the Feynman quote by @Stan Liou. This quote is being taken out of context, since Feynman is talking about the ways he views $mathematics$, which do not necessarily apply to physics. Indeed math and physics are related, but that does not mean they are classifiable identically! As others have noted here, physics is driven by empirical consensus, and people who make theories of physics try to stay ahead of the empirical/experimental progress by speculating and sometimes by generalizing. IT is in this generalizing vein that physics is axiomatic. But, lots of iterative, provisional progress is usually required in a field of physics before reliable generalizations are available to make axiomatic theories. For example, renormalization and regularization were ad-hoc fixes to issues in physics theories, but over time they have been made mathematically rigorous and there exist axiomatic approaches to these now, too. I hope my answer helps to clarify the nuances in the, seemingly simple, question "is physics axiomatic and rigorous?"