Netflix Inc. has tapped former ABC Entertainment President Channing Dungey as a vice president of original content, the latest evidence of the growing rivalry between the streaming-video giant and ABC parent Walt Disney Co.

ABC said last month that Ms. Dungey was stepping down from her post as Disney looks to complete its acquisition of the bulk of 21st Century Fox Inc.’s entertainment operations. That deal, which is expected to close early next year, will result in new management at ABC and the rest of Disney’s television...