The theorem holds for semigroups as well (well, in the finite volume setting! in the infinite volume setting there are subtleties between two-sided and one-sided averages).
Assume $\mu$ is a $S$-invariant and ergodic probability measure, where $S$ is a semigroup inside a unipotent subgroup $U$. We will show that $supp(\mu)$ supports contains a full $U=\langle S\rangle$ orbit, that's enough. Pick some $x\in supp(\mu)$. Consider $\overline{S.x}=P$. For a generic point, $P=supp(\mu)$ by the ergodic theorem. Notice that $S.P\subset P$, moreover $S^{2}.P=S.P$, so this is an $S$-invariant subset. By ergodicity $\mu(S.P)=1$, but we also have that $\mu(P)=1$ as well, so $P=S.P$ up to a measure zero set, or in other words $S^{-1}.P=P$ as well, hence $P$ is invariant under $S\cup S^{-1}$, now you may apply the orbit closure theorem.
P.S. you might want to consider the paper by Nimish Shah - "Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements" where he discuss the move from Ratner's theorem to discrete subgroup actions!