It is commonly known that it is the electron degeneracy pressure that prevents the collapse of a white dwarf into a neutron star, and it is not the electromagnetic force. However, it is also widely accepted that all the forces in the universe are just manifestations of the four elementary forces in the universe. So can the degeneracy pressure, caused due to Pauli's Exclusion principle, be shown to be due to one of these forces? I have read some other questions in SE How does the Pauli exclusion principle create a force in degenerate matter?, Degeneracy Pressure, What is it?, and they do not answer my question fully. In one of the links, the degeneracy pressure is described as an 'entropic force', but as far as I understand, all entropic forces can be explained by the underlying fundamental forces. If the degeneracy pressure is something that is fundamentally different from the other forces, why isn't it classified as a separate force?
The degeneracy pressure is indeed due to one of the four fundamental forces, but it takes a bit of though to see why.
If you put fermions into a box then their energy levels are quantised into the usual energy levels for a particle in a 3D box. So the first two fermions go into the ground state, then next two into a higher energy state and so on. Adding more and more fermions requires more and more energy giving us some total energy for all the fermions. It's the variation of this total energy of all the fermions with the volume of the box that determines the pressure. Note that this argument has assumed nothing about the particles other than that they are fermions, so no charge is assumed and therefore none of the four fundamental forces have been invoked.
Whch is all very well, but we then have to ask what is confining the fermions to the box? That is where the fundamental force comes in, because it's the gravitational force that is confining the fermions. Without gravity our gas of fermions would obviously just expand indefinitely to zero density and pressure. So gravity is the force ultimately responsible for the electron degeneracy pressure.
A degeneracy force is always due to some form of confinement. For example in ordinary matter the exchange force arises because electrons are confined into atoms due to the electrostatic forces.