Let us try making your statements and questions more precise. First point: the uncertainty principle is a feature of the quantum mechanics framework, it does not apply to a particular particle but to conjugated observables, such as, but not only, position and momentum.
Second point, because of the first point, you cannot "apply" the uncertainty principle to vacuum. What people usually do is study simple models to try and learn generic features, specifically here I am referring to the harmonic oscillator where, as you may know, the lowest energy eigenstate is not 0 but the zero-point energy, characteristic to the oscillator. This is of course a consequence of non-commuting operators and is related but not directly to the vacuum.
Third one has to move on to quantum field theory (QFT) to be able to speak about vacuum in the sense you mention. It is within this framework which is appropriate for higher energies, where the ground state of the system of fields is generally called the vacuum and one can construct a Fock space, which is the proper structure to handle multi-particle systems and therefore also describes a state without any particles. Within QFT one can compute the analogue of the zero-point energy but it diverges given that fields actually have infinite degrees of freedom (although finite, per space-time point), this is usually addressed by the normal ordering prescription, which takes care of removing this constant energy background.
On top of that there is a related issue called vacuum polarization, which has to do with higher order (in usual Feynman perturbation theory) contributions to this energy. This higher order contributions can be pictured as bubble diagrams made from the field content of the theory. They generally diverge and must be controlled by some renormalization procedure.
Having clarified everything that seemed to be mixed in the question, let me finish by trying to answer your last question.
The uncertainty principle is as it is stated a "principle" meaning is something that was proposed and which makes part of current theories of physics. It has survived already for roughly hundred years for a reason, it works. It helps us understand reality and is consistent with experiments (very different to religion). Having said that, it does not have/need a mathematical proof. However, this does not mean that a deeper explanation coming from further understanding in physics in the future, won't be able to get it as a consequence of even simpler principles.