There is no %100 proof in science; at least not for good science. It's always a question of being the most accurate / descriptive / useful theory. For example, Newtonian gravity is 'true' to the extent that it is very effective in a huge diversity of situations. General Relativity (GR) includes all of the accuracies of Newtonian Gravity, and then also explains a huge variety of additional phenomenon where Newtonian Gravity fails. We think that there are places where GR is incomplete: when you need to also describe things on the quantum mechanical scales. There are also some quirks about the 'dark sector' (Dark Matter, and Dark Energy) that we don't really understand. But, for all intents-and-purposes, GR can satisfactorily explain all observed gravitational phenomenon, including a wide variety of 'Tests of GR' --- which, very importantly, no other theory is able to do.
At the same time, the description of gravity through General Relativity is intrinsically that of 'curved' spacetime. The 'metric' of GR is fundamentally, and inextricably, a description of the very geometry of 3+1 spacetime which, purely from that, describes all of the resulting gravitational dynamics. A GR description is effectively synonymous with a curved-space-time description. To my knowledge, this is also unique to GR. Thus, by demonstrating the accuracy of GR, the validity of viewing gravity as curved spacetime is demonstrated. As described in this first paragraph, this should still subject to the same interpretation that this is currently the best description of observable properties of the universe.
Tests of General Relativity (brief summary, see wikipedia for details)
- "Perihelion Precession of Mercury" (dynamics) : Mercury's orbit is not perfectly closed ellipses, but instead 'precesses' (rotates slightly). The exotic dynamical effects have also been observed by space missions, and in the dynamics of stars moving near the massive black-hole in our galactic center.
- "Deflection of light by the sun" (lensing) : the path of light is observed to be deflected by massive objects. This was observed in the light of single stars moving behind the sun, but has since been extended to examples of completely distorted or duplicated images of galaxies, or subtle statistical effects to large fields of distant galaxies. The cause of the deflection (as read by GR) is very literally curvature of spacetime --- causing all objects (even photons, without rest-mass) to be deflected.
- "Gravitational Redshift" (light travel) : The frequency of light is 'redshifted' as it changes it's depth in gravitational potential wells. Similarly, the delaying effects of warped spacetime have long been observed, and is a very important component of how GPS works.
- Binary Pulsar (Gravitational Waves) : The orbital decay of the 'Hulse-Taylor' Binary-Pulsar is consistent with the emission of gravitational waves to an incredible precision, and won a Nobel prize. Gravitational waves are, very literally, traveling ripples in space-time which are able to carry energy.
- Cosmology : The expansion of space, and especially inflation, fit very nicely and naturally into the GR context --- because it describes the space-time itself, explicitly, instead of only the objects inside of it. The only alternative explanations we have for these observations are extremely convoluted - and require many different tools for different regimes (i.e. changing mass/light-speed etc may kind-of explain expansion observations, but you need something else to explain homogeneity and yet another thing for the horizon or monopole problem, etc). This is another example of a dynamic, flexible spacetime being employed.
In the very near future we expect to directly detect gravitational waves using Pulsar Timing Arrays and ground-based Laser Interferometers. This would be a 'nail in the coffin' for interpretation of gravity as spacetime.