We have carried out a neutron scattering investigation of the static structure factor S(q2 D) (q2 D is the in-plane wave vector) in the two-dimensional spin S=1/2 square-lattice Heisenberg antiferromagnet Sr2CuO2Cl2. For the spin correlation length ξ we find quantitative agreement with Monte Carlo results over a wide range of temperature. The combined Sr2CuO2Cl2-Monte Carlo data, which cover the length scale from ≈1 to 200 lattice constants, are predicted without adjustable parameteres by renormalized classical theory for the quantum nonlinear sigma model. For the structure factor peak S(0), on the other hand, we find S(0)∼ξ2 for the reduced temperature range 0.16<T/2 πρs<0.36, whereas current theories predict that at low temperatures S(0)∼T2ξ2. This discrepancy has important implications for the interpretation of many derivative quantities such as NMR relaxation rates. In the ordered phase, we have measured the temperature dependence of the out-of-plane spin-wave gap. Its low-temperature value of 5.0 meV corresponds to an XY anisotropy JXY/J=1.4×10-4. From measurements of the sublattice mangetization we obtain β=0.22±0.01 for the order parameter exponent. This may either reflect tricricality as in La2CuO4, or it may indicate finite-size two-dimensional XY behavior as suggested by Bramwell and Holdsworth. As in the S=1 system K2NiF4, the gap energy in Sr2CuO2Cl2 scales linearly with the order parameter up to the Néel temperature. We also reanalyze static structure factor data for K2NiF4 using the exact low temperature result for the correlation length of Hasenfratz and Niedermayer and including the Ising anisotropy explicitly. Excellent agreement between experiment and theory is obtained for the correlation length, albeit with the spin-stiffness ρs reduced by ≈20% from the spin-wave value. As in Sr2CuO2Cl2 we find that S(0)∼ξ2 for the reduced temperature range 0.22<T/2 πρs<0.47.