Actually, no - Windows ME did not depend on MS-DOS.
For all the bad rap it got (some of which was deserved), ME bypassed real-mode MS-DOS (and its config files) at bootup, and loaded its own protected-mode drivers, etc. in IO.SYS. It also loaded the main registry hive only once, and parallelized PnP resolution, significantly improving boot-up time. It also incorporated Windows 2000's networking stack, and added support for UPnP.
In many ways, ME was a stepping stone, getting some users on less-capable hardware onto Windows 2000 class OS features, without requiring a complete hardware upgrade.