A recent hack of Obamacare enrollment records might result in a full-blown privacy investigation of the government agency that is responsible for the federal health-care exchange and serve as a wake-up call to the government.

The Obamacare breach, which exposed 75,000 enrollment records, also might cause hospitals and health insurers to begin to question the government’s commitment to the very health-care data security rules they created.

The Centers for Medicare & Medicaid Services indicated that problems occurred in the federally...