In this work, we discretize the Euler equations of gas dynamics in space using a continuous Galerkin finite element method and a monolithic convex limiting (MCL) strategy. For the backward Euler time stepping, we show that the resulting nonlinear system has an invariant domain preserving (IDP) solution. Our proof involves constructing a fixed-point iteration that meets the requirements of a Krasnoselskii-type theorem. Our iterative solver for the nonlinear discrete problem employs a more efficient fixed-point iteration. The matrix of the associated linear system is a robust low-order Jacobian approximation that exploits the homogeneity property of the flux function. The limited antidiffusive terms are treated explicitly. We use positivity preservation as a stopping criterion for nonlinear iterations. The first iteration yields the solution of a linearized semi-implicit problem. This solution possesses the discrete conservation property but is generally not IDP. Further iterations are performed if any non-IDP states are detected. To facilitate convergence to steady-state solutions, we perform adaptive explicit underrelaxation at the end of each pseudo-time step. The adaptive calculation of appropriate relaxation factors is based on approximate minimization of nodal entropy residuals. In applications to standard two-dimensional test problems, we observe robust convergence behavior for all CFL numbers.