I hope to see a nicer proof which works for all $d$, but in the meantime, here's something ham-handed showing that the answer is no when $d=2$.

The Vietoris-Rips complex at scale $\epsilon$ for a point set $X$ in $\mathbb{R}^d$ coincides with the Čech complex for a set of balls of radius $\epsilon/2$ centered at the points of $X$. This is homotopy equivalent to the union of those balls, by the nerve lemma. That union of balls is a subspace of $\mathbb{R}^d$, so if we can show that such subspaces cannot have any higher homology, then we win. Unfortunately, there are counterexamples when $d\geq3$ [due to Barratt and Milnor](http://www.ams.org/journals/proc/1962-013-02/S0002-9939-1962-0137110-9/home.html).

However, subspaces of $\mathbb{R}^2$ do not have $k$-homology for $k\geq 2$ by [the results of Zastrow](http://at.yorku.ca/i/d/e/b/11.htm) cited in [this question](https://mathoverflow.net/questions/189323/can-a-subset-of-the-plane-have-nontrivial-h-2-or-pi-2). Most likely there are easier ways to show this for unions of $\epsilon/2$-balls in $\mathbb{R}^2$. For instance, if they are all homotopy equivalent to separable 1D metric spaces, then [work of Curtis and Fort](https://www.jstor.org/stable/1970184) applies. Or, there could be something hands-on that I'm missing.