Talking about periodic signals does not require any knowledge of sines / Fourier analysis, nor reference to any physical systems. The no-nonense definition is just this:
A periodic signal with period 𝑇 is a function 𝑓 : ℝ → ℝ such that 𝑇 : ℝ+ is the unique smallest positive number for which
𝑓(𝑡 + 𝑇) = 𝑓(𝑡) ∀ 𝑡∈ℝ
(there is no 𝜏 < 𝑇 that would also always give 𝑓(𝑡 + 𝜏) = 𝑓(𝑡)).
(We need the restriction forbidding 𝜏 < 𝑇 because else any signal with period 𝑇 would also qualify as periodic with period 2·𝑇 or 3·𝑇 and so on.)
That readily allows us to also define the fundamental frequency as simply the reciprocal: 𝜈f = 1⁄𝑇.
As I promised, this does not yet make any reference whatsoever to sinusoidal components – in fact, it also applies to functions that cannot be represented by a Fourier series, such as
⎧ 0 if 𝑡 is integral
𝑓(𝑡) := ⎨
⎩ sin (1⁄1−cos(2·π·𝑡)) else
In this signal, you have in essence locally (close to 0, 1, 2 etc.) the ripple-frequency go up arbitrarily high, without decreasing its amplitude). This can't be constructed as just a superposition of constant-frequency sine parts. Thus it's also not really possible to talk of harmonics.
Now, such a “signal” fortunately can never actually turn up in the physical world – you'd basically have an ultraviolet catastrophe at each of the points where the frequency goes infinite. Actual physical signals, if they are periodic, can be decomposed into a superposition of sinusoidals, with the slowest-oscillating sharing the fundamental frequency of the signal. (That includes, of course, the audio example above, which is effectively bandlimited through Nyquist-sampling. Mind, this is for this signal not possible without hefty aliasing artifacts.)
And then, the answer to your question is yes, as topo morto explained: if a signal has no harmonics, then by definition of “harmonic”, the signal is actually just a single sinusoidal.