In this work, we consider the monodomain problem to simulate the electrical activity within the heart tissue. From a mathematical standpoint, it is modeled by a nonlinear parabolic partial differential equation coupled with an ordinary differential equation, whose numerical resolution is often challenging and computationally expensive. We employ the backward Euler scheme for the time discretization and the conforming $\mathbb{P}_p$ finite element method for the spatial discretization. We derive a guaranteed a posteriori error estimate on the error between the exact solution at the continuous level and the approximate solution that is valid at each time step and each iteration of the linearization solver. Our estimate, based on equilibrated flux reconstructions, also distinguishes the spatial and temporal components of the error. The spatial component further consists of the discretization error and the linearization error. At each time step, we propose an adaptive Newton algorithm that stops the iterations once the estimated linearization error estimator becomes sufficiently small compared with the discretization error estimator and therefore no longer significantly affects the overall error estimate. In addition, this adaptive procedure leads to a reduction of the total number of iterations while ensuring the accuracy of the solution. Numerical experiments are presented to demonstrate the effectiveness of the proposed approach.