Gravity is a force. Forces accelerate objects. As the final step of the syllogism, gravity can accelerate objects. Hence the answer to you first question may be as simple as the property that gravity has to accelerate objects and give them energy is the "property" that gravity is a force. The electric force can also accelerate objects, and the magnetic force, etc.
As to the general question of why forces accelerate objects, Newton said bodies move at a constant speed in a straight line unless acted upon by a force.
This is of little help as natural scientists at that time also recognized the definition of force is that which will deviate a body from moving in a straight line at constant speed. Hence, you can see the definition of force and Newton's first law is somewhat circular.
Newton proposed something like "Hypothesis non fingo" meaning he did not know the answers underlying these deep questions. He simply applied his laws of motion and universal gravitation and optics to compute orbits, discover light is composed of a mixture of frequencies, as a by-product, invent the calculus, and contribute to, the calculus of variations with the brachistochrone etc.
Not a bad achievement considering he avoided considering the "fundamental" questions, like why is gravity a force? or why do forces accelerate objects? why is gravity actually a field? If gravity is a field, why do fields accelerate objects. If particles are actually following a minimum interval path is a curved space-time continuum (a la Einstein), why do they do that. In short, it is always possible to eventually get to an unanswerable question.
One possible answer to your original question may be that our universe just behaves this way where gravity, higher electromagnetism, etc are forces, and there are or maybe other universes out there or long ago that have different rules. Perhaps, life itself or humans or physicists are excluded from evolving in those other universes that follow different rules.
