You have to interpret the question restrictively to get a reasonable answer-domain. If you include mathematics, there are too many to list. I will ignore any theorem which is typically taught in graduate or undergraduate classes, and the result also has to be relatively easy to make completely rigorous, and the proof will have to involve a deep qualitative idea.
Here is a very partial list, based on whim:
- The Hawking area theorem, because the theorem and the proof both provide exceptional physical insight. This is detailed here: Second Law of Black Hole Thermodynamics . The Penrose theorem on gravitational collapse is similarly physical and similarly essential. There are many other results in gravitational physics that are phrased as theorems, although perhaps not with so elegant a proof.
- In statistical physics, there is another beautiful proof I think is simple and elegant, and has deep consequences: this is the no-passing theorem for elastic depinning due to Alan Middleton. The theorem says that if you start two interfaces in a depinning model with one behind the other, then the one behind will never overtake the one in front. The reason is that, in depinning, the rule for moving forward is determined by elastic forces, and when two interfaces, A and B, collide at point x, and A is behind B everywhere else, it is easy to see that the elastic forces on A hold the point x on A back more than the elastic forces hold the point x on B back. So if A moves forward, then B moves forward at least as much, and A cannot overtake B. This is extremely significant, because it means that the transition from no-motion to motion in depinning models must be second order.
- In high energy physics, there is the Froissart theorem: in a theory with a mass-gap, the total cross section can only grow as the squared logarithm of the center of mass energy. This is an essnetial theorem for the development of physics, which is unfairly left out of the curriculum, because of its associations with S-matrix theory.
- In thermodynamics, there is the Onsager Reciprocity theorems for near-equilbrium transport coefficients with time-reversal. This is foundational, it won the nobel prize, and it is not in the curriculum.
- In fluid dynamics, there is Helmholtz's theorem on the advection of vorticity in inviscid flow.