Unique Games results provide very interesting barriers to results through semidefinite programming. Lovasz theta ($\vartheta(G)$) function is an incarnation of SDP.
Is UG conjecture true $\iff \vartheta(G)$ is best approximation number for independence number $\alpha(G)$ of classes of graphs whose $\alpha(G)$ computation do not have efficient procedure if $P\neq NP$?
(I am adding $P\neq NP$ to avoid cases where other techniques such as Haemer's Bound gives better approximation).
As far as I know (and can interpret your question) no such result is known. There are two reasons:
1) Generally unique games hardness results (as well as NP hardness result) do not yield "instance based" hardness. That is, the UG-hardness result have the following flavor - "if the unique games conjecture is true then for problem $P$ no algorithm can get a better approximation ratio than $\alpha$, where $\alpha$ is the ratio that the canonical SDP achieves on $P$"
2) While the theta function is an appealing algorithm, and yields a $\sqrt{n}$ approximation for independent set for random graphs, in the worst case its approximation ratio can be close to $n$ (am not sure what's the extreme example), which can of course be achieved trivially. In fact, Hastad has gave an NP-hardness result ruling out an $n^{1-\epsilon}$ approximation for independent set for every $\epsilon>0$. So, even without the unique games conjecture, we know that essentially there is no non-trivial worst case approximation algorithm for this problem.
It would be very nice if there was a natural class of graphs where the theta function yields, say, an $O(\sqrt{n})$ approximation, and it is NP or UG hard to do better. I don't know of any such result.
