“She wants higher taxes. Bigger government. A single-payer radical government takeover of health care,” he said. “And those are her moderate positions.”

Though Ms. Abrams has said she likes the idea of a single-payer health care system, her immediate goal, she says, is to expand Medicaid under the Affordable Care Act in Georgia, one of 17 states that have declined to do so. On Tuesday she said Medicaid expansion would stem the wave of rural hospital closings, and noted that Vice President Mike Pence had overseen Medicaid expansion when he was governor of Indiana.

“Unfortunately my opponent, Mr. Kemp, does not have a plan for health care, other than saying trust your insurance companies,” she said.

Mr. Kemp does have a health care plan, which would, among other things, shore up a tax credit for rural hospitals. But his Trump-inspired campaign has captured the most attention with primary ads in which Mr. Kemp says he would personally round up “criminal illegals” in his pickup truck.

Mr. Kemp called out Ms. Abrams on Tuesday for saying recently that the coming “blue wave” would include undocumented immigrants. She has described the comment as a broad message of support for everyone living in the state. But Mr. Kemp interpreted it as a call for undocumented immigrants to vote.

“Well, Ms. Abrams, as you know in a recent video, you called on illegals to vote for you in this election,” he said. “I was actually shocked; I had to watch that video twice. It clearly shows that you are asking for undocumented and documented folks to be part of your winning strategy. So my question is why are you encouraging people to break the law for you in this election?”

Ms. Abrams responded that she had never “asked for anyone who is not legally eligible to vote to be able to cast a ballot.”