Excitonic insulators (EIs) arise from the formation of bound electron–hole pairs (excitons)1,2 in semiconductors and provide a solid-state platform for quantum many-boson physics3,4,5,6,7,8. Strong exciton–exciton repulsion is expected to stabilize condensed superfluid and crystalline phases by suppressing both density and phase fluctuations8,9,10,11. Although spectroscopic signatures of EIs have been reported6,12,13,14, conclusive evidence for strongly correlated EI states has remained elusive. Here we demonstrate a strongly correlated two-dimensional (2D) EI ground state formed in transition metal dichalcogenide (TMD) semiconductor double layers. A quasi-equilibrium spatially indirect exciton fluid is created when the bias voltage applied between the two electrically isolated TMD layers is tuned to a range that populates bound electron–hole pairs, but not free electrons or holes15,16,17. Capacitance measurements show that the fluid is exciton-compressible but charge-incompressible—direct thermodynamic evidence of the EI. The fluid is also strongly correlated with a dimensionless exciton coupling constant exceeding 10. We construct an exciton phase diagram that reveals both the Mott transition and interaction-stabilized quasi-condensation. Our experiment paves the path for realizing exotic quantum phases of excitons8, as well as multi-terminal exciton circuitry for applications18,19,20.