Abbott said a private insurance company would contract with the state to cover a pool of children. A court could order parents to buy the private insurance based on their ability to pay. The court could order the insurance premiums withheld from a parent's paycheck.

Parents who provide proof of insurance could opt out.

Federal and state laws already require noncustodial parents to meet their children's health insurance needs, Abbott said.

"We are not talking about a new responsibility but a new mechanism to ensure more parents meet their obligation," he said.