Apache supervises tasks in the form of threads and subprocesses, including managing memory in the form of pools, so yes, it's doing some of what an OS would do. It isn't, on the other hand, providing abstractions to hardware, and that's what I think of when I hear "OS". It's not really a cut-and-dried thing though: modern web browsers are like 90% of the way to being full-blown OS's, even down to abstracting hardware in the case of things like WebGPU, but they still rely on a host OS to bridge the connection and do the actual bit-banging required to make it work.
Doesn't that apply to IDEs that implement the language server protocol, since they spawn language servers and communicate with them?
And servers like Apache that spawn child processes to handle requests?