Insertions and deletions (indels) in protein-coding genes are important sources of genetic variation. Their role in creating new proteins may be especially important after gene duplication. However, little is known about how indels affect the divergence of duplicate genes. We here study thousands of duplicate genes in five fish (teleost) species with completely sequenced genomes. The ancestor of these species has been subject to a fish-specific genome duplication (FSGD) event that occurred approximately 350 Ma. We find that duplicate genes contain at least 25% more indels than single-copy genes. These indels accumulated preferentially in the first 40 my after the FSGD. A lack of widespread asymmetric indel accumulation indicates that both members of a duplicate gene pair typically experience relaxed selection. Strikingly, we observe a 30–80% excess of deletions over insertions that is consistent for indels of various lengths and across the five genomes. We also find that indels preferentially accumulate inside loop regions of protein secondary structure and in regions where amino acids are exposed to solvent. We show that duplicate genes with high indel density also show high DNA sequence divergence. Indel density, but not amino acid divergence, can explain a large proportion of the tertiary structure divergence between proteins encoded by duplicate genes. Our observations are consistent across all five fish species. Taken together, they suggest a general pattern of duplicate gene evolution in which indels are important driving forces of evolutionary change.