Newton's third law is that if a body A exerts a force on body B, then body B exerts an equal (in magnitude) and opposite (in direction) force on body A. In Newtonian gravity when body A exerts a gravitational force on body B, then body B exerts an equal gravitational force on body A. As both forces are attractive they have opposite directions.
General relativity doesn't treat gravity as a force, but Newton's third law can be demonstrated in the Newtonian limit of GR.
In big bang cosmology the initial expansion is not due to gravity or any other "force", rather it comes from the initial conditions of the Universe. Matter and radiation have positive energy density and non-negative pressure which both contribute to their gravity which acts to slow down the rate of expansion (the Hubble parameter). However it has been demonstrated that the Universe's rate of expansion is in fact increasing which means that there must exist some form of dark energy which has a negative pressure which is large enough in proportion to its (positive) energy density to accelerate the expansion of the Universe.
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