As per the universal law of attraction, any two bodies (having some mass) experience a force of 'attraction' which is proportionate to ...and ...inverse proportionate ....
Then comes my question: Why it should be force should be of type 'attraction' only ? Why it should not be repulsion / any other kind of force ?
To expand your quote concerning the gravitational force into an equation:
$$F_G = -\frac{Gm_1m_2}{r^2}$$
The force of gravity, $F_G$ is proportional to the product of the masses and inversely proportional to the distance, $r$, squared. Let's break this down and see what might cause $F_G$ to be positive.
In this equation, $r$ cannot be negative because it's a distance between two locations. Two locations cannot be a negative distance apart. And even if they somehow were, the squared would take care of that anyway.
$G$ is the universal constant and always positive. You might argue that it could possibly be negative, but that's not possible. $G$ actually doesn't really exist. It doesn't describe anything fundamental to the physics of the universe. $G$ is simply a bookkeeping constant that allows us to get the right answer for the force based on any choice of units for mass and distance. Technically, if one uses the "correct" units for mass and distance (e.g., the Planck units), then $G=1$ and effectively doesn't exist. Since $G$ is just a scaling factor that depends on the choice of units, it will only be a positive number.
That leaves us with the masses. These are the only things which could possibly be negative. Of course, to get a positive, repulsive force, one mass would have to be positive and the other negative. But what exactly is a negative mass? Mass is the metric which describes "how much" of something there is. How can you have less than nothing of something?
If you want to look at this another way, you can show that if mass could be negative, you'd get nonsensical results! Assuming of course, all other aspects of physics were the same. Recall from Newton's second law that
$$F = ma$$
Let's say there are two blocks sitting on a table. One block has a mass $m_1>0$ which is positive and the other has a mass $m_2<0$ which is negative. Ignore all other forces on these two blocks for the moment.
I go up to $m_1$ and I apply a force to push this mass forward. The acceleration that is induced is: $a = F/m_1$. Necessarily, the direction in which $m_1$ moves is the same direction in which I'm pushing. That's all well and good.
Now I go over to $m_2$ and I apply the same force, attempting to push it forward on the table. The acceleration induced on $m_2$ will be: $a = -F/|m_2|$. Note I made $m_2$ positive and pulled out the negative sign. You can see that if my force is forward, the direction the mass moves will be backwards! But here's the problem, my hand is in the way because it's trying to push to mass. As the mass tries to move backwards into my hand, it will be applying a force back on my hand, which by Newton's third law, necessarily mean's my hand is applying more force on the block, which then applies more force on my hand, ... and suddenly infinite forces are being applied or equivalently, these objects are infinitely accelerating. This is described by the concept of Runaway Motion.
If this seems strange to you, that's because it is. If negative masses existed, we'd live in a very weird universe. Fortunately, we live in a universe where physics makes sense, mass is positive, and by extension gravity is always attractive.
Why is gravity only an attractive force?
TL;DR
Because mass is always positive.
There are different notions of mass, but they're equivalent.
There are two distinct notions of mass: gravitational and inertial. The masses in Newton's law of gravitation, $F = \frac{Gm_1m_2}{r^2}$, are gravitational masses. The mass in Newton's second law of motion, $F=ma$, is inertial mass. Gravitational and inertial mass are implicitly assumed to be the same in Newtonian mechanics. General relativity makes this assumption explicit in the equivalence principle.
But what if they're not equivalent?
Unlike mathematics, where one can simply make an assumption and see where it leads, assumptions in physics need to be validated. This assumption has been tested with many kinds of materials, both on the ground and in space. Variations on the Cavendish experiment using different kinds of materials have been made. Within the limits of the rather lousy accuracy of the gravitational constant (one part per ten thousand, at best), every one of these is consistent with the null hypothesis (gravitational and inertial mass are the same) and inconsistent with the hypothesis that different materials have measurably different gravitational and inertial masses.
The Earth's Moon, with its very different near-side and far-side, provides an even better mechanism for testing this equivalence. Rather than the one part per ten thousand (at best) accuracy available to Cavendish-style experiments, the Moon shows that gravitational and inertial mass for sodium and iron are equivalent to within about one part per ten trillion.
So much for ordinary matter, but what about antimatter?
That an ordinary matter particle and its antimatter equivalent have the same (positive) inertial mass has been tested over and over in particle colliders around the world. Whether the equivalence principle also applies to antimatter remains a somewhat open question. While there are many reasons to think that the equivalence principle applies to antimatter as well as normal matter, testing that this is the case is very hard. The best results to date are from the ALPHA experiment, which tests whether neutral antihydrogen (a antiproton and an positron) falls up or down. The results are that antihydrogen's gravitational mass lies somewhere between -65 and 120 times its inertial mass. This is not anywhere close to conclusive, but it does lean towards antimatter having a positive gravitational mass, consistent with the equivalence principle.
Along the same lines with previous answers suggesting "mass cannot be negative," I'd like to add an insight for why that might probably be the case. If Higgs field and particles' varying degrees of interaction with the field is what gives rise to what we call mass, then the theory suggests that photons don't have mass (and constitute the velocity limit through space) because they don't interact with the field at all. I don't think the framework allows for negative interaction with the field or an "anti-Higgs" field.
Theoretically, gravity can be "attractive" in the sense that objects move towards you when pushed. This can occur from negative mass (doesn't seem to make sense, but theoretically possible). Peter Engels and others have written a paper about it here and it's an interesting idea.
The idea is that by cooling the atoms to almost absolute zero, they create a Bose-Einstein condensate and act likes waves in the realm of quantum dynamics.
