No, they are not etymologically related. The early development of stable homotopy theory happened simultaneously with the early developments of scheme theory, so certainly neither terminology was influenced by the other.
Grothendieck's choice of terminology of the "spectrum" of a ring comes from functional analysis. One can speak of the eigenvalue spectrum of an operator, which can be generalized to the spectrum of a whole family of mutually commuting operators, which can be abstracted to the (Gelfand) spectrum of a commutative $C^\ast$-algebra. The Gelfand-Naimark theorem says that the $C^\ast$-algebra is canonically the algebra of functions on the topological space given by the spectrum, just as in scheme theory.
Spectra in topology were introduced by Lima and his advisor Spanier. It is a minor mystery why they chose the name "spectrum", but most likely it is used in the second sense of this definition:
(a) An array of entities, as light waves or particles, ordered in accordance with the magnitudes of a common physical property, as wavelength or mass: often the band of colors produced when sunlight is passed through a prism, comprising red, orange, yellow, green, blue, indigo, and violet. (b) This band or series of colors together with extensions at the ends that are not visible to the eye, but that can be studied by means of photography, heat effects, etc., and that are produced by the dispersion of radiant energy other than ordinary light rays. Compare band spectrum, electromagnetic spectrum, mass spectrum.
- A broad range of varied but related ideas or objects, the individual features of which tend to overlap so as to form a continuous series or sequence: the spectrum of political beliefs.