Shouldn't the infinite sea of electrons contribute to gravity? According to my understanding of the dirac equation, there's an infinite sea of electrons occupying all negative energy states which prevents an electron from dropping to lower and lower energy states down to negative infinity.
Since these are electrons, they have obviously have a charge, and hence every electron sits in the potential of every other: essentially there is a collosal amount of energy in all of space from these sea electrons.
Shouldn't this energy contribute to gravity? I understand that the cosmological constant should be energy present at all points in space, which causes space to expand. Wouldn't this energy do the same?
 A: Do not confuse models with reality/data.
The hypothesis of infinite seas of electrons has been left behind because better mathematical models were developed, and the problems of real electrons in infinite seas are model problems. 
The solutions of the Dirac equation described fermions for us and started the way to quantum field theory which is the present tool of studying particle physics.
In the standard model of particle physics, all the  point elementary particles in the table cover all of space as fields, on which operators creating and annihilating the particles act. Thus charged massive  particles exist only where the creation operators act, the dirac model is not applicable. What is kept from the dirac equation is the plane wave solution for fermions,(the maxwell for photons,...) which represents the quantum  field of the particle at all points in space, i.e. the electron field  in your case.
The vacuum expectation value for all these fields is zero, except for the Higgs boson.
(It is important to keep in mind that this is the quantum framework, where wavefunctions lead to probabilities of interaction, not certitudes.) 
