WASHINGTON—The Labor Department extended minimum-wage and overtime pay to nearly two million workers in the home health-care industry, in a long-fought victory for unions, but a move some business officials said could make home care unaffordable for many consumers.

The workers—often known as personal-care aides, home-health aides or certified nursing assistants—typically bathe, dress and feed elderly or disabled patients. A large percentage of the workers are hired directly by people with disabilities or their families. Others...