What is the equation connecting gravitational force and specific gravity?

There is no direct equation connecting Gravitational force and Specific gravity.
Gravitational force is the force with which two bodies of mass ##m_1## and ##m_2## attract each other. If one of the bodies is earth the expression reduces to
##F_Gravity=G(M_e m)/R_e^2##
where ##G## is Universal Gravitational Constant ##M_e and R_e## are the mass and radius of earth respectively and ##m## is the mass of other object.
This equation can be written as first three are constants
##F_Gravity=mgN##
here ##g## is the local due to gravity.
##N## newton being the unit of force. ##F_Gravity## is the weight of body.
Specific gravity is the ratio of the density of a substance to the density of a reference substance.
For solids and liquids reference substance is water at ##4^@C## and at one atmospheric .
For gases it is air at room temperature ##21^@C## and at one atmospheric pressure.
We know that the density ##rho-=mass per unit volume## of the substance under test.
We can therefore write
##Specific Gravity = frac {rho_text{sample}}{rho_{ H_2O}} ##
We see that specific gravity is a dimensionless quantity as it is a ratio of two densities.
Indirectly both are related as follows
Specific gravity can be computed from the expression for weights ##W## of sample and water both of equal volume ##V##
##SG = frac {rho_text{sample}}{rho_(H_2O}} = frac {(m_text{sample}//V)}{(m_{ H_2O}//V)}##
## = frac {m_text{sample}}{m_{ H_2O}} ##
Multiplying and dividing with ##g##
## = frac {m_text{sample}}{m_{ H_2O}} g/g##
or ##SG= frac {W_{V_text{sample}}}{W_{V_{ H_2O}}} ##