Graham's comments follow the president's reported remarks last week about "shithole" countries.
"I said he was a xenophobic, race-baiting religious bigot. I ran out of things to say. He won."
Senator Lindsey Graham criticizes CNN for calling Trump a "kook," but he used the same word to insult the president a year ago.
The Russian president denied involvement in the U.S. election.
Lindsey Graham suggests Trump should focus on helping people rather than settling scores or lobbing insults.
Graham represents South Carolina, which holds a key early vote in presidential contest.
Carson's remarks came after the real estate mogul went on a lengthy rant about him.
The four candidates dueled over policy differences on immigration, foreign policy and Donald Trump.
During a CNN debate between Republican presidential candidates, John McCain's daughter took to social media to show some love.
Low-polling GOP candidates that included South Carolina Senator Lindsey Graham and Louisiana Governor Bobby Jindal sparred over everything from foreign policy to immigration.
The 80-minute debate featured seven Republican candidates whose polling numbers were too low to qualify for the prime-time event on Fox News.
The second-tier Republican presidential candidates had trouble editing down their Hillary critiques.
Graham admonished Trump, who currently leads in the polls among Republicans, to "stop being a jackass."
Graham, from South Carolina, is the ninth Republican to declare he is running for the White House.
Graham said existing U.S. legislation "would cut off aid to the Palestinians if they filed a complaint" against Israel.