In this paper, we develop an adaptive version of the inexact Uzawa algorithm applied to finite element discretizations of the linear Stokes problem. We base our developments on an equilibrated flux a posteriori error estimate distinguishing the different error components, namely the discretization error component, the inner algebraic solver error component, and the outer Uzawa iteration error component. On each outer Uzawa and inner linear algebraic solver iteration, we prove that our estimate gives a guaranteed upper bound on the total error, as well as a polynomial-degree-robust local efficiency. Our adaptive inexact algorithm stops the outer Uzawa iteration and the inner linear algebraic solver iteration when the Uzawa error component, respectively the algebraic solver error component, do not have a significant influence on the total error. The developed framework covers all standard conforming and conforming stabilized finite element methods. The implementation into the FreeFem++ programming language is invoked and two numerical examples showcase the performance of our adaptive strategy.