Universal power laws have been scrutinised in physics and beyond, and a long-standing debate exists in econophysics regarding the strict universality of the nonlinear price impact, commonly referred to as the square-root law (SRL). The SRL posits that the average price impact III follows a power law with respect to transaction volume QQQ, such that I(Q)∝QδI(Q) \propto Q^{\delta}I(Q)∝Qδ with δ≈1/2\delta \approx 1/2δ≈1/2. Some researchers argue that the exponent δ\deltaδ should be system-specific, without universality. Conversely, others contend that δ\deltaδ should be exactly 1/21/21/2 for all stocks across all countries, implying universality. However, resolving this debate requires high-precision measurements of δ\deltaδ with errors of around 0.1 across hundreds of stocks, which has been extremely challenging due to the scarcity of large microscopic datasets—those that enable tracking the trading behaviour of all individual accounts. Here we conclusively support the universality hypothesis of the SRL by a complete survey of all trading accounts for all liquid stocks on the Tokyo Stock Exchange (TSE) over eight years. Using this comprehensive microscopic dataset, we show that the exponent δ\deltaδ is equal to 1/21/21/2 within statistical errors at both the individual stock level and the individual trader level. Additionally, we rejected two prominent models supporting the nonuniversality hypothesis: the Gabaix–Gopikrishnan–Plerou–Stanley and the Farmer–Gerig–Lillo–Waelbroeck models. Our work provides exceptionally high-precision evidence for the universality hypothesis in social science and could prove useful in evaluating the price impact by large investors—an important topic even among practitioners.