To the question of the title I would "intuitively" answer yes, by the following informal argument:
Covariance "measures the strength of linear association" (when scaled by the product of standard deviations) between two variables, while Spearman's rho "measures the strength of monotone association."
Linear association is a subset of monotone association (isn't it?), hence, when the measure of monotone association is zero, the measure of linear association should also be zero.
But I have learned my lesson (and so I am not a menace to society) about easy "intuitive" arguments in Statistics. And my attempts to examine this conjecture formally were not fruitful so far.
So: Does a zero Spearman's rho imply zero Covariance?
Can we formally prove it, or disprove it even by a counter-example?
This post provides also examples that there is no such relation