No. Some 8-bit computers had interpreted languages in ROM despite not having anything reasonably resembling a modern operating system. The Apple 2 is one example. You could boot the system without any disks or tapes, and it would go straight to a BASIC prompt, where you could write basic (no pun intended) programs.
Note that an operating system is somewhat of a vague term when speaking about these days - these 8-bit computers did have some level of firmware, and this firmware did provide some OS-type functionality like access to basic peripherals. In these days, what we now know as an OS was more commonly called a "DOS" - a Disk Operating System. MS-DOS is one of them, as well as Apple's ProDOS. These DOS's evolved into our modern-day operating systems (e.g. Windows 95 was based on top of MS-DOS, while modern Windows versions derive from a separate branch that was largely re-implemented with more modern techniques), so one could claim that their ancestors are the closest they had to what we now call an OS.
But what is an interpreter but a piece of software?
In a more theoretical sense, an interpreter is simply software - a program that takes input and produces output. Suppose you were to implement a custom solid-state Turing Machine. In this case, your "input" would be the program to be interpreted, and the "output" would be the program's behavior. If "software" can run without an operating system, then an interpreter can.
Is this model a little simplified? Of course. The difference is a matter of degree, not nature. Add very basic user input and output capabilities (e.g. a TTY) and you have the foundation to implement all, or nearly all, of the basic functionality of a language such as Java byte code, Python, or BASIC. The main things you would be missing are libraries and whatnot that depend on things like screen manipulation, multiprocessing, and networking, but you could handle them with time too.