All six cases involved in the lawsuit concern female same-sex couples who have decided to have a child or children.
The agency said it will work quickly to make sure that its offices and employees are ready to deal with the changes.
Paxton adds that while officials are allowed to choose not to issue same-sex marriage licenses, they may face litigation and/or a fine for doing so.
Several dozen people gathered within earshot of the governor's office in the Statehouse rotunda less than two hours after the ruling came down.
INSIDE: A collection of statements from Indiana leaders, organizations and politicians about SCOTUS' decision on same-sex marriage.
In a landmark 5-4 decision, the U.S. Supreme Court overturned bans on same-sex marriage. Civil rights activists are celebrating the victory in the courts.
By a 5-4 decision, the U.S. Supreme Court has reversed the sixth circuit’s decision and ends bans against same-sex marriage.
The high court is expected to rule by the end of the month on whether states can limit marriage to heterosexual couples.
More than 1,100 Indiana same-sex couples have obtained marriage licenses this year following court 2014 court rulings that struck down the state's gay-marriage ban.
19 men and 12 women whose same-sex marriage cases from Ohio and Kentucky, plus Michigan and Tennessee, will be argued at the Supreme Court on April 28.