The static and dynamic spin fluctuations in the spin S=1, two-dimensional (2D) square-lattice antiferromagnet La2NiO4 have been studied over a wide temperature range using neutron scattering techniques. The spin correlations in La2NiO4 exhibit a crossover from two- to three-dimensional (3D) behavior as the Néel temperature is approached from above. Critical slowing down of the low-energy spin fluctuations is also observed just above TN. The correlation length, ξ(T), and the static structure factor, S(0), have been measured and are compared with recent theoretical calculations for the quantum 2D Heisenberg antiferromagnet using microscopic parameters determined from previous spin-wave measurements. Good agreement for ξ(T) is found with the exact low-temperature result of Hasenfratz and Niedermeyer provided that 2 πps is renormalized by ≈20% from the spin-wave value.