Adsorption of self-avoiding tethered membranes of hexagonal orientation on a solid impenetrable plane is studied by means of Monte Carlo computer simulations of a coarse-grained continuum model, varying the membrane linear size L and the strength epsilon of the short-range attractive wall potential. A second-order adsorption transition is found to take place at a critical strength of the adsorption potential epsilon(c), as predicted earlier for binding manifolds in the so-called strong fluctuating regime. By means of finite-size scaling analysis for membranes of size 5<or=L<or=30, containing 61<or=N<or=2611 monomers, we find that the critical crossover exponent for adsorption phi approximately 0.60+/-0.01. Thus the fraction m of adsorbed segments at epsilon(c) is found to scale as m proportional to N(phi-1)=N(-0.4). The membrane thickness lambda(min) decreases with growing strength epsilon of the adhesive potential as lambda(min) proportional to mid R:epsilon/epsilon(c)-1mid R:(-psi), where psi approximately 0.58+/-0.02. The monomer density profiles of adsorbed membranes decay exponentially with the distance z from the substrate rho(z) proportional to exp[-(z/xi)], where the correlation length xi proportional to mid R:epsilon/epsilon(c)-1mid R:(-psi) with psi approximately 0.70+/-0.01. We also investigate the kinetics of adsorption of a polymerized membrane in the regime of strong adsorption and find that the order parameter variation with elapsed time during the adsorption process is given by a power law m(t) proportional to t(omega), where omega approximately 1.0, regardless of the strength of the adsorbing potential epsilon>>epsilon(c). The characteristic time for complete adsorption in this regime scales as tau(ads) proportional to L(2). Regarding the strength epsilon of the substrate potential, tau(ads) is found to diminish linearly as the respective equilibrium value of the order parameter increases. A simple analytic model yields also tau(ads) proportional to L(2) while suggesting that the process of adsorption cannot be identified by simple "unrolling."