We define the log-gamma sheet and the log-gamma landscape in terms of the 2-parameter and 4-parameter free energy of the log-gamma polymer model and prove that they converge to the Airy sheet and the directed landscape, which are central objects in the Kardar-Parisi-Zhang (KPZ) universality class. Our proof of convergence to the Airy sheet relies on the invariance of free energy through the geometric RSK correspondence and the monotonicity of the free energy. To upgrade the convergence to the directed landscape, tail bounds in both spatial and temporal directions are required. However, due to the lack of scaling invariance in the discrete log-gamma polymer--unlike the Brownian setting of the O'Connell-Yor model--existing on-diagonal fluctuation bounds are insufficient. We therefore develop new off-diagonal local fluctuation estimates for the log-gamma polymer.