Why do ionic compounds dissociate whereas coordinate complexes won't? An ionic bond is the bonding between a non-metal and a metal, that occurs when charged atoms (ions) attract after one loses one or more of its electrons,and gives it to the other molecule, for example sodium and chlorine. This makes the bond stronger and harder to break.
  In other words, an ionic bond is the electrostatic force of attraction between two oppositely charged ions. The positive ion is called cation, and the negative ion is the anion. It is like the north and south poles of a magnet.  
  A covalent bond is a chemical bond that involves the sharing of electron pairs between atoms. The stable balance of attractive and repulsive forces between atoms when they share electrons is known as covalent bonding. For many molecules, the sharing of electrons allows each atom to attain the equivalent of a full outer shell, corresponding to a stable electronic configuration.  
  A dipolar bond, also known as a dative covalent bond or coordinate bond is a kind of 2-center, 2-electron covalent bond in which the two electrons derive from the same atom.  
  NOTE: It is important to recognize that pure ionic bonding - in which one atom "steals" an electron from another - cannot exist: all ionic compounds have some degree of covalent bonding, or electron sharing. Thus, the term "ionic bond" is given to a bond in which the ionic character is greater than the covalent character - that is, a bond in which a large electronegativity difference exists between the two atoms, causing the bond to be more polar (ionic) than other forms of covalent bonding where electrons are shared more equally. Bonds with partially ionic and partially covalent character are called polar covalent bonds. Nevertheless, ionic bonding is considered to be a form of noncovalent bonding.
  


$$\begin{equation}\tag{1}F=\frac{1}{4\pi\epsilon}\cdot\frac{q_1q_2}{r^2}\end{equation}  $$
 According to Coulomb's law equation $(1)$, the force between two charges depend on the medium in which they are placed. The value of $\epsilon_r$ (relative permitivity) of air is found to be approx $1$, whereas relative permittivity of water is found to approx $80$ at room temperature.   

I thought Coulombs law could answer why ionic compounds dissociate into ions whereas other compounds of different bonds won't. I thought, ionic compounds are formed by the electrostatic interaction of two oppositely charged ions (I assumed these to be charges), they will be stable in air, as relative permittivity value is not much effective. When, the same is dissolved in water, I thought they will dissociate as relative permittivity value of water effects more. I found the explanation, so as to why ionic compounds dissociate into ions when dissolved in water. In case of coordinate bond, I thought an atom losses a pair of electron and thus can be called as cation, whereas the other becomes anion. So, even here we can expect electrostatic interaction, and as discussed above must dissociate into ions when dissolved in water. But coordination complex of coordinate compound won't dissociate? I think, I have misunderstood somewhere or else there must be some chemistry theory which could explain, why coordinate complex won't dissociate?  

Even in case of covalent bond we can see the bond to be formed by stable electrostatic attraction and repulsion between electrons and protons, so I thought, even covalent bonds should break, when dissolved in water, as electrostatic interaction is going to be affected. I thought, there must be some other forces which must be keeping them bonded. If so what are they?  

I don't know whether I am right or wrong, if any was the case please explain.
First of all, who says coordination complexes don't dissociate? The interesting chemistry happens right there, and without dissociation no catalytic coordination compounds would be possible!

Now, the way I see it, when you move your compound, be it ionic, covalent or coordination, from air to a condensed phase, it will automatically interact with said phase. I think that you assume this phase is water, but it might as well be dichloromethane or ammonia or diethyl ether or anything else that is liquid (basically).

What makes a bond happen (or not) is the overall energy of the system. See, in chemistry (as in physics) the energy minimum is the goal of the system, since it is the most stable. Using this line of reasoning we can derive that bonds are formed because the overall energy (of the bonding electrons) is lower than if they were not bonded. Note that it doesn't matter where these electrons come from (they might be shared by both bonding partners or made available by only one of them). A good and easy way to visualize this is if you scratch the surface of Molecular Orbital Theory.

So why do ionic compounds dissolve? Mostly due to the solvent-compound interactions. See more here.

(I also think that your approach using Coulomb theory doesn't work that well, since the change of the dielectric constant around the compounds has nothing to do with what is between the bonding partners, thus affecting attractive forces between the two partners.)