Percolation theory is applied to the phase-transition dynamics of domain pattern formation in segregating binary Bose--Einstein condensates in quasi-two-dimensional systems. Our finite-size-scaling analysis shows that the percolation threshold of the initial domain pattern emerging from the dynamic instability is close to 0.5 for strongly repulsive condensates. The percolation probability is universally described with a scaling function when the probability is rescaled by the characteristic domain size in the dynamic scaling regime of the phase-ordering kinetics, independent of the intercomponent interaction. It is revealed that an infinite domain wall sandwiched between percolating domains in the two condensates has an noninteger fractal dimension and keeps the scaling behavior during the dynamic scaling regime. This result seems to be in contrast to the argument that the dynamic scale invariance is violated in the presence of an infinite topological defect in numerical cosmology.
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