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Does there exist a complex smooth proper variety whose fundamental group is finite non-abelian simple?

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Yes. In fact, Serre proved that any finite group is the fundamental group of a smooth projective complex variety. See Proposition 15 of:

J.-P. Serre, Sur la topologie des variétés algébriques en charactéristique $p$, Symposium Internacional de Topologia Algebraica, Universidad Nacional Autonoma de Mexico, 1958, pp. 24–53 (MSN).

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  • $\begingroup$ The proof of this fact can also be found in Shafarevich's Basic Algebraic Geometry 2 (third edition), chapter 9, section 4.2. Another reference is Fundamental Groups of Compact Kähler Manifolds by Amorós et al., example 1.11. $\endgroup$
    – Michael Albanese
    Commented Jun 20, 2021 at 22:08

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