Blazars constitute the vast majority of extragalactic $gamma$-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 multiwavelength follow-up of very high-energy neutrino events of plausible astrophysical origin. Spatial and temporal coincidences of these neutrino events with the high-activity state of $gamma$-ray blazars can provide a unique opportunity to decipher cosmic-ray interactions in the relativistic jets. Assuming that blazars accelerate cosmic rays up to ultrahigh energies ($E>10^{17}$ eV), we calculate the guaranteed contribution to the line-of-sight cosmogenic $gamma$-ray and neutrino fluxes from four blazars associated with IceCube neutrino events. Detection of these fluxes by upcoming $gamma$-ray imaging telescopes like CTA and/or by planned neutrino detectors like IceCube-Gen2 may lead to the first direct signature(s) of ultrahigh-energy cosmic-ray (UHECR) sources. 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 $gtrsim 10$ times the inferred neutrino luminosity from the associated IceCube events. Blazars TXS~0506+056, 3HSP~J095507.9+355101 and GB6~J1040+0617 can be detected by CTA if the UHECR luminosity is $gtrsim 10$ times the neutrino luminosity inferred from the associated IceCube events. 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 and $gamma$-ray telescopes.
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