Bauxite residue, known as “red mud,” is a potential raw material for extracting rare-earth elements (REEs). The main REEs (Sc, Y, La, Ce, Nd, Nb, and Sm) from the raw bauxite are concentrated in RM after the Bayer leaching process. The earlier worldwide studies were focused on the scandium (Sc) extraction from RM by concentrated acids to enhance the extraction degree. This leads to the dissolution of major oxides (Fe2O3 and Al2O3) from RM. This article studies the possibility of selective Sc extraction from alkali fusion red mud (RMF) by diluted nitric acid (HNO3) leaching at pH ≥ 2 to prevent co-dissolution of Fe2O3. RMF samples were analyzed by X-ray fluorescence spectrometry (XRF), X-ray diffraction (XRD), electron probe microanalysis (EPMA), and inductively coupled plasma mass spectrometry (ICP-MS). It was revealed that Sc concentration in RMF can reach up to 140–150 mg kg−1. Sc extraction was 71.2% at RMF leaching by HNO3 at pH 2 and 80 °C during 90 min. The leaching solution contained 8 mg L−1 Sc and a high amount of other REEs in the presence of relatively low concentrations of impurity elements such as Fe, Al, Ti, Ca, etc. The kinetic analysis of experimental data by the shrinking core model showed that Sc leaching process is limited by the interfacial diffusion and the diffusion through the product layer. The apparent activation energy (Ea) was 19.5 kJ/mol. The linear dependence of Sc extraction on magnesium (Mg) extraction was revealed. According to EPMA of RMF, Sc is associated with iron minerals rather than Mg. This allows us to conclude that Mg acts as a leaching agent for the extraction of Sc presented in the RMF in an ion-exchangeable phase.