Finite subgroups are always contained in the maximal compact subgroup so if $ G $ has a finite maximal closed subgroup then $ G $ must be compact. Also a maximal subgroup always includes the center so if $ G $ has a finite simple subgroup which maximal then $ G $ must have trivial center in addition to being compact. Thus the only groups which have finite maximal closed subgroups are adjoint groups like $ SO(2n+1), PSO(2n),PU(n) $
Note that a finite simple group $
\Gamma $ is a maximal closed subgroup of $ PU_n $ if and only if the central extension $ n.G $ is a unitary $ 2 $ design as a subgroup of $ SU_n $. See Claim 3 of https://math.stackexchange.com/a/4477296/758507
Some examples of finite simple groups appearing as subgroups of $ PU_n $ are given here
https://mathoverflow.net/questions/414265/alternating-subgroups-of-mathrmsu-n
here
https://mathoverflow.net/questions/414315/finite-simple-groups-and-mathrmsu-n
the references in
https://mathoverflow.net/questions/17072/the-finite-subgroups-of-sun
and here
https://mathoverflow.net/questions/344218/on-the-finite-simple-groups-with-an-irreducible-complex-representation-of-a-give
But when it comes to the maximality of such finite simple subgroups of $ PU_n $ the most useful reference is
https://arxiv.org/abs/1810.02507
which, read correctly, supplies a full classification of maximal closed subgroups of $ PU_d $ that happen to be finite and simple.
The classification consists of a few infinite families of examples of maximal closed subgroups of $ PU_d $ which are finite and simple
$ PU_d $, $ d=\frac{3^k -1}{2} $ and $ d=\frac{3^k +1}{2} $ both have a maximal $ PSp_{2k}(3) $ for $ k \geq 2 $.
$ PU_d $, $ d=\frac{2^k-(-1)^k}{3} $ has a maximal $ PSU_k(2) $ for $ k \geq 4 $
In addition to these, there are a few dimensions $ d $ for which $ PU_d $ has more maximal closed finite simple subgroups than we would expect. These exceptional case are:
$ PU_2 $: $ A_5 $
$ PU_3 $: $ A_6,GL_3(2) $
$ PU_4 $: $ A_7,PSU_4(2) $
$ PU_6 $: $ A_7,PSL_3(4), PSU_4(3) $
$ PU_8 $: $ PSL_3(4) $
$ PU_{10} $: $ M_{11}, M_{12} $
$ PU_{12} $: $ Suz $
$ PU_{14} $: $ ^2 B_2(8) $
$PU_{18} $: $ J_3 $
$PU_{26} $: $ ^3 F_4(2)' $
$PU_{28} $: $ Ru $
$PU_{45} $: $ M_{23},M_{24} $
$PU_{342}$: $ O'N $
$PU_{1333}$: $ J_4 $
All these finite simple maximal closed subgroups of $ PU_n $ lift to finite quasisimple maximal closed subgroups. Some of these quasi simple lifts have simple "section" so to speak and thus correspond to a finite simple maximal closed subgroups of $ SU_n $. Some examples are
$ SU_3 $: $ GL_3(2) $
$ SU_6 $: $ A_7 $
Best Answer
The image of a finite subgroup of $\text{SU}(2)$ in $\text{SO}(3)$ is a finite subgroup of $\text{SO}(3)$; moreover, the kernel is either trivial or $\{ \pm 1 \}$. But $-1$ is the unique element of order $2$ in $\text{SU}(2)$, so any group of even order contains it.
I claim all the finite subgroups of odd order are cyclic. This follows because the inclusion $G \to \text{SU}(2)$ cannot define an irreducible representation of $G$ (since otherwise $2 | |G|$), hence it must break up into a direct sum of dual $1$-dimensional representations.
So once you know the finite subgroups of $\text{SO}(3)$, you already know the finite subgroups of $\text{SU}(2)$.