Exoplanetary systems host giant planets on substantially noncircular, close-in orbits. We propose that these eccentricities arise in a phase of giant impacts, analogous to the final stage of solar system assembly that formed Earth’s Moon. In this scenario, the planets scatter each other and collide, with corresponding mass growth as they merge. We numerically integrate an ensemble of systems with varying total planet mass, allowing for collisional growth, to show that (1) the high-eccentricity giants observed today may have formed preferentially in systems of higher initial total planet mass, and (2) the upper bound on the observed giant planet eccentricity distribution is consistent with planet-planet scattering. We predict that mergers will produce a population of high-mass giant planets between 1 and 8 au from their stars.