We can explain ionic bond as the force between charged particles due to Coulomb electrostatic law. This got me wondering how is then covalent bond explained purely in terms of physics?
Eventually, every kind of bond in condensed matter is reducible to electrostatics. Covalent bond, ionic bond, van der Waals forces, metallic interactions and even more sophisticated interactions like the phonon-mediated electron-electron attraction in a superconductor, are all reducible to purely electrostatic interactions between electrons and nuclei.
What ensures the observed variety of behaviors is the fact that at least one component of the matter does not behave like classical particles but QM is required, in addition to the introduction of effective interactions between a reduced number of degrees of freedom.
Covalent bonding is a quantum-mechanical phenomenon that can't be explained classically. The basic idea can be demonstrated in the case of a hydrogen molecule if you think of the electrons as two particles in a box. By joining the two hydrogen atoms together into a molecule, you make the "box" twice as long. Let's pretend that this is a rectangular box. Lengthening the cubical box into a $2\times 1 \times 1$ box makes the wavelength along the long axis twice as big. By the de Broglie relation $p=h/\lambda$, this reduces the momentum in that direction by a factor of $2$, which makes the associated kinetic energy $1/4$ as much. The lowered energy makes the system more stable. To pull it apart, you would need to add energy.