In this study, we propose a new spectral clustering framework that can auto-tune the parameters of the clustering algorithm in the context of speaker diarization. The proposed framework uses normalized maximum eigengap (NME) values to estimate the number of clusters and the parameters for the threshold of the elements of each row in an affinity matrix during spectral clustering, without the use of parameter tuning on the development set. Even through this hands-off approach, we achieve a comparable or better performance across various evaluation sets than the results found using traditional clustering methods that apply careful parameter tuning and development data. A relative improvement of 17% in the speaker error rate on the well-known CALLHOME evaluation set shows the effectiveness of our proposed spectral clustering with auto-tuning.