Asynchronous calls don't even need to occur on the same system/device as the one invoking the call. So if the question is, does an asynchronous call require a thread in the current process, the answer is no. However, there must be a thread of execution somewhere processing the asynchronous request.
Thread of execution is a vague term. In a cooperative tasking systems such as the early Macintosh and Windows OS'es, the thread of execution could simply be the same process that made the request running another stack, instruction pointer, etc... However, when people generally talk about asynchronous calls, they typically mean calls that are handled by another thread if it is intra-process (i.e. within the same process) or by another process if it is inter-process.
Note that inter-process (or interprocess) communication (IPC) is commonly generalized to include intra-process communication, since the techniques for locking, and synchronizing data are usually the same regardless of what process the separate threads of execution run in.