So Newton's universal law of gravitation is:
$$F = GMm/r^2.$$
However, if I wanted to solve for distance using Newton's law of gravitation, the equation would be:
$$r^2 = GMm/F.$$
Thus if I wanted to solve for $r$, I would square root both sides. But this would lead to two solutions, positive and negative $r$. I would really like to know either if my calculations are wrong or how negative distance would make sense.
In the expression $$|\vec F|=G\frac{Mm}{r^2}$$ the quantity $r$ is the distance between the two bodies, and is non-negative by definition. Therefore, if you rearranged this expression to isolate $r$ you would choose the positive square root and disregard the negative root.
So Newton's universal law of gravitation is:
F = GMm/r^2
A better form for Newton's universal law of gravitation is $$\vec F = -\frac{GMm}{r^2}\hat r$$ where $r^2=\vec r \cdot \vec r$ is shorthand for the dot product of $\vec r$ with itself and $\hat r = \vec r/|\vec r|$ is the unit vector in the direction of $\vec r$.
With this, then expanding the above law for $\vec F = \left( F_x, F_y, F_z\right)$ and $\vec r = \left( r_x, r_y, r_z \right)$ gives $$\left(F_x,F_y,F_z\right)=-GMm\left( \frac{r_x}{\left( r_x^2+r_y^2+r_z^2 \right)^{3/2}},\frac{r_y}{\left( r_x^2+r_y^2+r_z^2 \right)^{3/2}},\frac{r_z}{\left( r_x^2+r_y^2+r_z^2 \right)^{3/2}} \right)$$ which is 3 equations in 3 unknowns and can be solved uniquely for any non-zero $\vec F$.
