Because only infinite dimensional, unitary representation of $SL(2,F)$ for a local field $F$ can fail to be tempered, if they are spherical. This follows from the classification. This is true for $GL(2)$ as well modulo twisting by one-dimensional representations, which are trivial for $SL(2)$. I think Wallach "Real reductive groups vol 1+2" and Knapp "Representation theory of semisimple groups" covers this for $\mathbb{R}$ and $\mathbb{C}$.
As Asaf points out, looking at the K=SU(2)-invariant vectors will do the job.
Getting to the finite-volume setting: You will obtain Eisenstein series and cusp forms and constant functions. There you actually have non-tempered representations besides the trivial representation. There are atmost finitely many. For $\Gamma$ a congruence subgroup of an imaginary quadratic field $k$, it is an important conjecture that there are non (the analogue of the Selberg eigenvalue conjecture). Then you actually have a much bigger group acting, i.e., $SL_2(A_k)$ via strong approximation, i.e. there exists an open subgroup$K_\Gamma$ of $SL_2(A_{k,f})$ (finite adeles) $$ \Gamma \backslash SL_2(\mathbb{C}) = SL_2(k) \backslash SL_2(A_k) / K_\Gamma.$$
If you consider an irreducible representation in there, it factors into representations of $SL_2(k_v)$ for each place as a tensor product, all but finitely many are spherical. It is assumed that they should all be tempered (besides the one-dimensional representations). This is know as the Ramanujan Peterson conjecture. Non trivial bounds are known due to Blomer and Brumley (they actually work with $GL(2)$). Here is survey: Blomer + Brumley -The role of the Ramanujan conjecture in analytic number theory, Bulletin AMS 50 (2013), 267-320
Usually the decomposition is $V_{cusp} + V_{const} + V_{cont}$, where $V_{cusp}$ are the cuspidal representations, $V_{const}$ the one-dimensional representation and $V_{cont}$ is the continuous representation. $V_{cont}$ is known to be tempered and can be explicitly given. $V_{cusp}$ is fairly unkown and only very few are conjectured to be related to Galoisrepresentations, which would imply temperedness automatically.