DC distribution systems can provide effective solutions for integrating renewable energy sources in future power systems. The low inertia of the DC distribution system causes several problems regarding stability and fault response. For stability, this low inertia represents a major cause for instability and voltage oscillations, especially with constant power loads. For fault response, the low inertia results in very high fault currents with a significant rate of rising and limited damping. This study aims to use superconducting fault current limiter (SFCL) as a virtual inertia for DC distribution systems under various disturbances and fault conditions. The system description and modeling are first presented, including the detailed dynamics of SFCL. The stability analysis for the DC system is carried out using the Hurwitz criterion, from which the suitable range of SFCL resistance is identified. Considering this range and fault current limitation, an SFCL resistance of $1.5~\Omega $ is adopted. The whole system is implemented using PSCAD/EMTDC software. A series of case studies are investigated to validate the effectiveness of SFCL in strengthening the inertia of the DC system. These case studies include sudden variations in supply voltage, sudden load changes, and faults. SFCL could successfully suppress voltage oscillations, keep voltage stability, and limit fault currents.