String-localized quantum field theory allows renormalizable couplings involving massive vector bosons, without invoking negative-norm states and compensating ghosts. We analyze the most general coupling of a massive vector boson to a scalar field, and find that the scalar field necessarily comes with a quartic potential which has the precise shape of the shifted Higgs potential. In other words: the shape of the Higgs potential has not to be assumed, but arises as a consistency condition. We derive this result as a consequence of the "Principle of String-Independence" (PSI) for renormalized perturbation theory with string-localized interactions: While the renormalizable interaction density may be localized along an auxiliary "string", the S-matrix (and also the local observables of the theory) must not depend on it. Along the way, we develop a model-independent scheme how the PSI is implemented in string-localized perturbation theory, and how it can be used as a tool to "renormalize the non-renormalizable".