Virtual synchronous generator (VSG) could provide virtual damping and inertia for an islanded microgrid, which enhances the system frequency and voltage supporting capability. However, the additional introduced virtual inertia in VSG increases the system order, which aggravates the oscillation possibility of the output active power during the transient states and even may endanger the system stability. In this study, a novel fractional-order model predictive controller (FOMPC) is proposed for a fractional-order virtual synchronous generator controller (FOVSG) to alleviate the output power oscillation and achieve an optimal frequency and voltage regulation for an islanded microgrid. First, the FOVSG model is established by changing the integer-order virtual inertia into the fractional-order. Second, based on the definition of the Grunwald-Letnikov (GL) fractional calculus, the integral-order model predictive control (MPC) is extended to the FOMPC. Then a cost function is designed for frequency deviation and rated power variation of FOMPC-FOVSG. Finally, the optimal control law is obtained by solving a quadratic programming problem. Hardware-in-the-loop (HIL) experiments were carried out to demonstrate the superiority of FOMPC-FOVSG over existing VSG techniques under several scenarios.