Yes. The $\sigma(A^{**},A^*)$-closure of $I$ in the second dual von Neumann algebra $A^{**}$ is an ultraweakly closed right ideal, which is of the form $pA^{**}$ for some projection $p$. (In fact $p$ is the ultrastrong limit of any left approximate unit of $I$.) Thus,

$$\sup_{x\in I,\ \|x\|=1}\| xa \| = \sup_{x\in A^{**},\ \|x\|=1}\| pxa \|$$
for every $a\in A^{**}$. Consider the central support projection $z$ of $p$ in $A^{**}$. Since $Ia$ is nonzero for every nonzero $a\in A$, the $*$-homomorphism $A\ni a\mapsto za \in zA^{**}$ is faithful. Now, let $a\in A$ be given. For any $\epsilon>0$, the projection $q:=z1_{[\|a\|-\epsilon,\|a\|]}(|aa^*|^{1/2})$ in $zA^{**}$ is nonzero and so there is a nonzero partial isometry $v$ in $zA^{**}$ such that $v=pv=vq$. It follows that
$$\sup_{x\in A^{**},\ \|x\|=1}\| pxa \| \geq \| pva \| \geq \|a\|-\epsilon.$$
Since $\epsilon>0$ was arbitrary, we are done.

Another proof is to use Kadison's transitivity theorem in combination with the following fact: if $J$ is an essential ideal in $A$, then $\|a \|=\sup_\pi\|\pi(a)\|$, where $\pi$ runs over irreducible $*$-representations of $A$ which do not kill $J$.