One of the primary goals of Hayabusa2 is to land on the asteroid Ryugu to collect its surface materials. The key for a successful touchdown is to find a promising landing site that meets both scientific and engineering requirements. Due to the limited availability of pre-arrival information about Ryugu, the landing site selection (LSS) must be conducted based on proximity observations over a limited length of time. In addition, Ryugu was discovered to possess an unexpectedly high abundance of boulders with an absence of wide and flat areas, further complicating the LSS. To resolve these problems, we developed a systematic and stepwise LSS process with a focus on the surface topography of Ryugu and the associated touchdown safety. The proposed LSS scheme consists of two phases: Phase-I LSS, a comprehensive survey of potential landing areas at the 100-m scale based on the global mapping of Ryugu, and Phase-II LSS, a narrowing-down process of the candidate landing sites at the 10-m scale using high-resolution images and a local terrain model. To verify the feasibility of a precision landing at the target site, we also investigated the landing dispersion via a Monte Carlo simulation, which incorporates the effect of the irregular surface gravity field. One of the major characteristics of the Hayabusa2 LSS developed in this study is the iterative feedback between LSS analyses on the ground and actual spacecraft operations near the target asteroid. Using the newly developed method, we chose a landing site with a radius of 3 m, and Hayabusa2 successfully conducted its first touchdown on February 21, 2019. This paper reports the methodology and results of the stepwise iterative LSS for the first Hayabusa2 touchdown. The touchdown operation results reconstructed from flight data are also provided, demonstrating the validity of the adopted LSS strategy.