To stop issuing Civil Marriage licenses does not change all the laws in the State where Civil Marriage is a component. They may not allow new marriages, but they would have to do exensive re-writing of a large chunk of state laws. Inheretiance, Tax law, parental responsibilities, divorce, child custody, etc.
Secondly, just because it stops issuing Civil Marriage licenses doen't mean that they can stop recognizing Civil Marriages in other states as long as they recognize Civil Marriges for state residents.
WW