Context.Blazars constitute the vast majority of extragalacticγ-ray sources. They can also contribute a sizable fraction of the diffuse astrophysical neutrinos detected by IceCube. In the past few years, the real-time alert system of IceCube has led to the multiwavelength follow-up of very high-energy neutrino events of plausible astrophysical origin. Spatial and temporal coincidences of a number of these neutrino events withγ-ray blazars provide a unique opportunity to decipher cosmic-ray interactions in the relativistic jets.Aims.The aim of this work is to test if theγ-ray blazars associated with the IceCube neutrino events are also sources of ultra-high-energy cosmic rays (UHECRs;E > 1018eV).Methods.Assuming that blazars accelerate UHECRs, we calculate the “guaranteed” contribution to the line-of-sight cosmogenicγ-ray and neutrino fluxes from four blazars associated with IceCube neutrino events. We compare these fluxes with the sensitivities of the upcomingγ-ray imaging telescopes, such as the CTA, and with the planned neutrino detectors, such as IceCube-Gen2.Results.We find that detection of the cosmogenic neutrino fluxes from the blazars TXS 0506+056, PKS 1502+106, and GB6 J1040+0617 would require UHECR luminosity ≳10 times the inferred neutrino luminosity from the associated IceCube events, with the maximum UHECR proton energyEp, max ≈ 1020eV. Cosmogenicγ-ray emission from blazars TXS 0506+056, 3HSP J095507.9 +355101, and GB6 J1040+0617 can be detected by the CTA if the UHECR luminosity is ≳10 times the neutrino luminosity inferred from the associated IceCube events and forEp, max ≳ 1019eV.Conclusions.Detection of cosmogenic neutrino and/orγ-ray flux(es) from blazars associated with IceCube neutrinos may lead to the first direct signature(s) of UHECR sources. Given their relatively low redshifts and hence total energetics, TXS 0506+056 and 3HSP J095507.9+355101 should be the prime targets for upcoming large neutrino andγ-ray telescopes.