Yes, it does imply that, but it is a stronger assumption: Setting the conditional mean of the error terms to zero is a stronger assumption than just setting their marginal mean to zero. If we have $\mathbb{E}(U | X = x) = 0$ for all allowable $x$ then using the law-of-iterated-expectations we get:
$$\mathbb{E}(U) = \mathbb{E}(\mathbb{E}(U | X)) = \mathbb{E}(0) = 0.$$
In fact, we can also derive another useful property from this assumption:
$$\begin{equation} \begin{aligned}
\mathbb{Cov}(U,X)
&= \mathbb{E}(UX) - \mathbb{E}(U)\mathbb{E}(X) \\[6pt]
&= \mathbb{E}(X \cdot\mathbb{E}(U | X)) - \mathbb{E}(U)\mathbb{E}(X) \\[6pt]
&= \mathbb{E}(X \cdot 0) - 0 \cdot\mathbb{E}(X) = 0.
\end{aligned} \end{equation}$$
This tells use that the conditional mean assumption implies that the marginal mean of the errors is zero, and the error terms are also uncorrelated with the explanatory variables.
Note that this is a stronger assumption than if we just assume that $\mathbb{E}(U) = 0$. That is, you can have situations where the error terms have a marginal mean of zero, even though they are correlated with the explanatory variable $X$.