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If we take into account the retarded potentials and the motion of the Sun(due to the planet), does Newton's Gravitational theory give rise to the motion of Perihelion of planets (qualitatively, not experimental correctness) ?

Edit : Consider case with only 2 bodies. One very massive .Consider only point particles

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It seems to me that one can answer this from classical electromagnetism, where all the requirements of relativity, including retarded potentials, are respected. In this case one finds that there is a precession of the axes of a quasi-elliptical orbit in a $1/r$ potential, but this is associated with velocity and is given (angle change per orbit) by $$ \delta \phi = \frac{\pi \alpha^2}{L^2 c^2} $$ where the force is ${\bf f} = - \alpha \hat{\bf r}/r^2$ and $L$ is the angular momentum, and the formula applies in the case $L \gg \alpha/c$. This precession is about 6 times smaller than the one predicted by General Relativity for gravitating masses.

In this result one assumes the central mass is large and not moving, and one simply assumes the $1/r^2$ force without calculating potentials etc. But a full calculation of Coulomb interaction, including retarded potentials, does indeed give a $1/r^2$ force in the case of charges at rest, so we can apply the above result in the limit $v/c \rightarrow 0$. That means the minimum radius of the orbit should be large. This precession effect is not, I would say, particularly to do with the concept of retardation in the potential. It is related to the fact that the momentum is not proportional to velocity. Almost any change away from the Newtonian case with $1/r$ potential will lead to precession, because in the Newtonian case the absence of precession relies on a (mathematical) symmetry which will no longer hold if one changes the dynamical equation.

(Of course precession effects in the solar system are dominated by other things, especially the effect of Jupiter).

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