Modular multilevel converter (MMC) has been widely used as critical interfaces between the grid and large-scale renewable energy sources (RESs). Due to its large number of submodule (SM) capacitors, MMC can store substantial amounts of energy, offering potential for inertia support. However, existing research lacks a comprehensive method for assessing the maximum inertia of MMC, resulting in an unclear safe operating region and suboptimal utilization of its inertia support capability. This article introduces a precise method for assessing the maximum inertia of MMC by first deriving the permissible range of SM capacitor voltage, followed by an accurate energy-to-inertia conversion method that remains unaffected by capacitor voltage variations. Building on this, an adaptive inertia support control (AISC) strategy is developed, which maximizes the MMC's inertia under various operating conditions and adaptively switches the operating modes based on the rate of change of frequency (RoCoF) to enhance the stability of the dc link voltage. Hardware-in-the-loop (HIL) verification results demonstrate that the inertia provided by AISC in both modes aligns with the assessment, reducing RoCoF by 65.5% and 70.7%, and the maximum frequency deviation by 30.6% and 34.2%, respectively.