I am claiming head of household. My adult dependent did not obtain health insurance, Income was $0. Can he obtain an exemption to avoid me having to paying penalty?

[CarolynM]

No, by claiming him as a dependent, you are accepting the responsibility for providing health insurance or paying the penalty.

If is not you dependent and had no income, he would fall under the income threshold and not pay the penalty. 

For more information, follow these links:

https://turbotax.intuit.com/tax-tips/health-care/health-care-and-your-taxes-whats-the-connection/L7p... 

ACA Exemptions 

Healthcare Penalty Estimator