(Rewritten to give an answer more useful to future visitors.)

First of all, as noted in comments, there is no (countably additive) Gaussian measure on $H$ with covariance operator the identity.

However, if we take $\gamma$ to be some other Gaussian measure, the answer is no, $H^*$ is not dense in $L^2(H, \gamma)$. One way to see this is that every $f \in H^*$, considered as a random variable on $(H, \gamma)$, has a centered Gaussian distribution (in which we include the "degenerate" Gaussian distribution which is the constant 0). In particular $f$ has mean zero, and the mean-zero random variables are a proper closed subspace of $L^2(\gamma)$. (So for instance, the nonzero constants are not in the closure of $H^*$.)

Indeed, $L^2$ limit of centered Gaussian random variables is again centered Gaussian (this actually holds if you replace "$L^2$" by "in distribution"). Thus every random variable in the $L^2$ closure of $H^*$ is centered Gaussian, and hence non-Gaussian $L^2$ functions on $H$ are not in the closure of $H^*$ either.

It is true that if you allow *polynomials* $F(x) = p(f_1(x), \dots, f_n(x))$ where $f_1, \dots, f_n \in H^*$, then all such functions $F$ are in $L^2(H,\gamma)$, and they form a dense subspace. Indeed, if you let $p$ range over all Hermite polynomials of degree $n$, then the closed span of all corresponding $F$ gives you the space $\mathcal{H}_n$ which is the $n$th Wiener chaos, and we have an orthogonal decomposition $L^2(H, \gamma) = \bigoplus \mathcal{H}_n$. These spaces are also the eigenspaces of the Ornstein-Uhlenbeck operator $N$ (aka number operator).

Back to the "measure" with covariance operator $I$: we can consider such a "measure" as a finitely additive measure on the cylinder sets of $H$. I suppose it might be possible to study an $L^2$ space with respect to this finitely additive measure. I don't know much about such spaces, but I would guess that a similar argument would show that $H^*$ is still not dense.