Adopting high penetration levels of electric vehicles (EVs) necessitates the implementation of appropriate charging management systems to mitigate their negative impacts on power distribution networks. Currently, most of the proposed EV charging management techniques rely on the availability of high-bandwidth communication links. Such techniques are far from realization due to ① the lack of utility-grade communication systems in many cases such as secondary (low-voltage) power distribution systems to which EVs are connected, rural areas, remote communities, and islands, and ② existing fears and concerns about the data privacy of EV users and cyber-physical security. For these cases, appropriate local control schemes are needed to ensure the adequate management of EV charging without violating the grid operation requirements. Accordingly, this paper introduces a new communication-less management strategy for EV charging in droop-controlled islanded microgrids. The proposed strategy is autonomous, as it is based on the measurement of system frequency and local bus voltages. The proposed strategy implements a social charging fairness policy during periods when the microgrid distributed generators (DGs) are in short supply by allocating more system capacity to the EVs with less charging in the past. Furthermore, a novel communication-less EV load shedding scheme is incorporated into the management strategy to provide relief to the microgrid during events of severe undervoltage or underfrequency occurrences due to factors such as high loading or DG outages. Numerical simulations demonstrate the superiority of the proposed strategy over the state-of-the-art controllers in modulating the EV charging demand to counteract microgrid instability.