Comment: "You are required to have car insurance to drive on our public roads."

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Wrong. You are required to have LIABILITY insurance as a condition of being granted a driver's license. This protects other motorists and your own assets. (Insurance on the vehicle is usually a condition of borrowing the money to buy the car. This protects the lien holder, no one else, although it's prudent to protect your own assets.) 

To apply the parallel to health coverage, you would require insurance as a condition of being alive. This is unique in the entire economy. (Never mind that you have the option of paying for your health care by numerous other means.)

Huge distinction.

While it's a wise purchase -- like auto or homeowners insurance -- health insurance doesn't necessarily protect anyone else. If your interest is enforcing responsibility -- as defined by whom, by the way? -- one is simply liable for his own bills. That requires no mandate or enforcement by the state over a private transaction.