Is the expected utility the inverse of the utility function?

This is trivially not true. Consider simple example of utility:

$$u(x) = x^{1/2}$$

Expected utility $E(u(x)) = E[x^{1/2}]$

Inverse utility is $u^{-1} implies x = u^2 $

clearly generally $E(u) neq u^{-1} $.