The tool used specifically to monitor the free space in the vCenter Server database is vimtop. Vimtop is a performance monitoring utility that provides detailed information about the running processes and resource usage on vCenter Server, making it suitable for monitoring the database's free space. Esxtop, df, and vmstat are tools and commands for monitoring system performance and disk usage on ESXi hosts or Unix systems, but they are not tailored specifically for vCenter Server database monitoring.