"I'm looking forward to a September debate" with DeMint, Greene said of the effort to educate voters. "I would like an hour debate live on one of the networks."
"I will also need the party's backing with funding on the state level and national level," he said.
It's unclear how much money, if any, the party will give to Greene, whom many political analysts don't give a chance against DeMint.
"You had an absolute unknown [Greene] running against a virtual unknown [Rawl]," said Winthrop University political scientist Scott Huffmon. Greene's victory "says something about the depth of the Democratic bench in South Carolina, but not much more than that."
Huffmon says DeMint and Republicans have never been concerned with either potential challenger ahead of the general election and said conspiracy theories that Republicans may have facilitated Greene's victory to give DeMint a weaker challenger are misguided.
Greene's victory could have negative implications for the Democratic Party, however, since Greene could appear to be a "sacrificial lamb." "If the party doesn't put time and energy into helping someone unknown get respectable turn out, then it could look very bad for the party," he said.
ABC News' Luis Martinez, Michael Murray and Gregory Simmons contributed to this report.