Newton limit is approximation of GR in weak fields and SMALL velocities.
Small velocities means, that whole 4-velocity of a particle is basically in time component. So you can imagine, that if spacetime is curved same in all directions, than the time component is most significant simply because the particle almost doesn't move in space at all.
To be more precise:
The spacetime around spherically symmetrical field is given by Schwarzschild metric (in natural units):
$$
ds^2=-\left(1-\frac{r_s}{r}\right)dt^2+\left(1-\frac{r_s}{r}\right)^{-1}dr^2+r^2d\Omega\approx ds^2_{flat}+\frac{r_s}{r}(dt^2+dr^2)
$$
where $r_s$ is Schwarzschild radius and $ds^2_{flat}$ is Minkowski part (flat spacetime part) of the metric.
As you clearly see, the perturbation of flat spacetime metric has same magnitude in time component as in space component in natural units.
But now, let us compute geodesics. The geodesics equation is given by:
$$
a^\mu=-\Gamma^\mu_{\alpha\beta}v^{\alpha} v^{\beta}
$$
where $a^\mu$ is 4-acceleration of a particles, $v^\mu$ its 4-velocity and $\Gamma^\mu_{\alpha\beta}$ is Christoffel symbol. Now, the relevant Christoffel symbols for radial motion are $\Gamma^t_{\alpha\beta}$ and $\Gamma^r_{\alpha\beta}$ of which nonzero are only:
$$
\Gamma^t_{tr}=\Gamma^t_{rt}\approx -g_{tt,r}/2
$$
$$
\Gamma^r_{rr}\approx g_{rr,r}/2
$$
$$
\Gamma^r_{tt}\approx -g_{tt,r}/2
$$
and all of them are of same order since perturbations of metric components $g_{tt}$ and $g_{rr}$ are of same order (in fact $g_{tt,r}=g_{rr,r}$). So the geodesic equation for radial motion in weak field of spherically symmetric source is:
$$
a^t=-\Gamma^t_{\alpha\beta}v^{\alpha} v^{\beta}\approx g_{tt,r}v^{t} v^{r}
$$
$$
a^r=-\Gamma^r_{\alpha\beta}v^{\alpha} v^{\beta}\approx g_{tt,r}v^{t} v^{t}/2-g_{rr,r}v^{r} v^{r}/2=g_{tt,r}/2
$$
Where I have used $g_{tt,r}=g_{rr,r}$ from the metric and $v^{t}v^{t}-v^{r}v^{r}=1$ from normalization.
Having 4-acceleration we can get radial 3-acceleration component ($a^r_3$) using:
$$a^r=a^t v^r/\gamma+\gamma^2 a^r_3$$
where $\gamma$ is Lorentz factor.
Now this doesn't lead to Newtonian gravitation law without assumption, that velocities are small. With this assumption $\gamma\approx 1$, $v^t\approx-1$, $v^r\ll 1$ and $v^\mu\approx (-1,\vec{v})$ and the equation simplifies further:
$$a^r\approx a^t v^r+a^r_3 => a^r_3 \approx a^r - a^t v^r$$
Substituting from geodesic equation:
$$
a^r_3\approx g_{tt,r}/2 - g_{tt,r}v^{t} (v^{r})^2=g_{tt,r}/2+o((v^{r})^2)\approx r_s/(2r^2)=-GM/r^2
$$
with $M$ being mass of the source, as Newton gravitation says. So the approximation is not that space-components of curvature can be neglected, it is in the fact that space-components of 4-velocity can be neglected.
