Bacteriophage T4 gene 32 protein (gp32) is a single-stranded DNA (ssDNA) binding protein, and is essential for DNA replication, recombination and repair. While gp32 binds preferentially and cooperatively to ssDNA, it has not been observed to lower the thermal melting temperature of natural double-stranded DNA (dsDNA). However, in single-molecule stretching experiments, gp32 significantly destabilizes lambda DNA. In this study, we develop a theory of the effect of the protein on single dsDNA stretching curves, and apply it to the measured dependence of the DNA overstretching force on pulling rate in the presence of the full-length and two truncated forms of the protein. This allows us to calculate the rate of cooperative growth of single clusters of protein along ssDNA that are formed as the dsDNA molecule is stretched, as well as determine the site size of the protein binding to ssDNA. The rate of cooperative binding (ka) of both gp32 and of its proteolytic fragment *I (which lacks 48 residues from the C terminus) varies non-linearly with protein concentration, and appears to exceed the diffusion limit. We develop a model of protein association with the ends of growing clusters of cooperatively bound protein enhanced by 1-D diffusion along dsDNA, under the condition of protein excess. Upon globally fitting ka versus protein concentration, we determine the binding site size and the non-cooperative binding constants to dsDNA for gp32 and I. Our experiment mimics the growth of clusters of gp32 that likely exist at the DNA replication fork in vivo, and explains the origin of the "kinetic block" to dsDNA melting by gene 32 protein observed in thermal melting experiments.