The study of the origin of irregular satellites remains important in planetary science because it provides constraints on the formation process of giant planets and probes the properties of a now-extinct planetesimal disk that existed at 5-30 AU early in solar system history. While several putative scenarios of irregular-satellite capture around giant planets have been developed, various uncertainties and the lack of an accurate model of the evolutionary history of the solar system usually prevent an assessment of their overall likelihood. Here we study a three-body interaction scenario in which irregular satellites are formed by dissociation of a planetesimal binary in the gravity field of a planet. Within the frame of the Nice model, we determine how many irregular satellites are expected to be formed about each of the giant planets. We pay special attention to a possible capture of Triton via this mechanism. We find that Triton could have been captured via a binary dissociation very soon after Neptune's formation when the planetesimal disk was still dynamically cold. Triton was most likely captured by a dissociation of a binary system where the more massive component was ~2-5 times heavier than Triton. Our results suggest that Neptune, the formation of Triton's binary, and the capture of Triton around Neptune all occurred within the first ~5-10 Myr of solar system formation when the gas disk was still present. This would rule out the late formation of ice giants. Our results also indicate that binary dissociation is a highly unlikely process for the origin of small irregular satellites for two reasons. First, the orbital distribution of the captured bodies is inconsistent with that of the observed irregular satellites. Second, the efficiency of the captures is too low to explain the numerous populations of small irregular satellites.