Why do objects with mass have gravitational force that is proportional to their mass? Why do objects with mass have gravitational force that is proportional to their mass, i.e the larger the object the more gravitational force it has?
 A: Why is the force of gravity proportional to the mass of the body exerting it?
Just because.
This answer may disappoint you, but it's the truth.  In  Newtonian physics, the equation $ F = G \frac{m_1   m_2}{r} $ is axiomatic, just like Newton's laws of motion. None of these  is derived from more fundamental propositions, either. 
So your question, while perfectly reasonable, is like asking,  "why does a body in motion stay in motion unless a force acts on it?" Or, "why is the force that it takes to accelerate the body proportional to its mass?". Or, "why does every action have an equal and opposite reaction?" The answer is the same for all these axioms: That's just the way it is in Newton's theory. There is no "why". The buck of asking "why" stops the moment you hit an axiom of a theory.
You tagged your question "Newtonian Physics", but I want to add that a relativistic approach to it would not make the answer any more interesting or insightful. In Einstein' General Relativity Theory, one body creates a curvature in space-time that's proportional to its mass. The other body then 'feels' a force that's proportional to the curvature of space-time, and to its own mass. Why? Just because. These, too,  are axioms, axioms of General Relativity theory in this case. so the buck stops, once again, before it ever gets going. 
A: The gravitational force an object experiences is proportional to its mass because of intimate ties between gravity and acceleration. The mass of an object influences the gravitational field but it does not determine how the object moves through the gravitational field. Suppose we fix a mass $M$ at the origin. This generates a gravitational field
$$g=GM/r^2$$
where $r$ is the distance from the origin and the field is directed towards the origin. If we place a test mass $m$ in this field the force is
$$F=mg=GMm/r^2$$
and it looks like the mass $m$ is important for determining the motion of the test mass. However, notice that when we want to determine the motion of the test mass we need the acceleration and $m$ drops out
$$a=F/m=GM/r^2.$$
In a deeper sense we can think about general relativity where the mass of an object determines the curvature of spacetime. However, the mass does not affect how the object moves through spacetime. Objects of different mass will just move in a straight line through curved spacetime without any regard for their mass differences (unless there is some other force, e.g., electromagnetic).
A: It's an empirical relation. Empirical relations are those relations which can't be derived but verified through experiments and observation. F=ma is also empirical. 
A: If we ignore general relativistic corrections, then gravity itself does not contribute to gravity. This means that the gravity from each part of the object adds up to the gravity from the other parts. The gravity of an object is therefore proportional to the mass of the object.
A: In short, it leads to fundamental question about gravity, which is not yet answered.
This in other words, is asking - what is the mechanism by which mass tells space how to curve, and space tells mass how to move. Both of these happen in proportion to amount of mass and energy.
Answer to "how much" is known (and it is in proportion to mass/energy), but answer to "how/why" is not known yet. I do not think scientists are really interested in knowing, because, answering it, will result into other unanswered questions - no end.
