ATLANTA—Since Georgians went to the polls in 2016, more than half a million voters have been registered, including an increase in the voting ranks of women and minorities—groups that tend to lean Democratic.

Much of that increase is the result of a change to driver’s-license forms enacted two years ago under outgoing Gov. Nathan Deal, a Republican. With the change, new applicants and renewals are automatically registered to vote unless they expressly opt out.

That...