"Few things are more central to our duty than confirming Supreme Court justices who will uphold the law and apply it equally for all Americans, not rewrite it from the bench based on personal opinion," DeMint said.
DeMint, a Greenville Republican, is not on the Senate Judiciary Committee, which will hold confirmation hearings for Sotomayor.
Sen. Lindsey Graham, a Seneca Republican who does sit on the judiciary panel, was more guarded in his response to Obama's choice of Sotomayor as the first Hispanic nominated for the high court.
"I do not know Judge Sotomayor," Graham said. "I look forward to meeting with her and discussing the important issues confronting the court."
Graham was among the Senate Judiciary Committee members Obama called last week to discuss his pending high court nomination. Graham was out of the country Tuesday, but his aides said they didn't believe Obama had named Sotomayor during their phone talk.
Graham said he intends "to be fair and firm in (his) questioning of the nominee" during the committee confirmation sessions.
"The hearings can be a valuable public service, as they give us a window into the nominee's philosophy and disposition," Graham said. "I hope we will have a meaningful opportunity to explore the qualifications, judicial temperament and judicial philosophy of Judge Sotomayor."
Republican President H.W. Bush appointed Sotomayor to the federal district bench in 1992. Democrat Bill Clinton elevated her to the 2nd U.S. Circuit Court of Appeals.
After an extensive delay over political disputes, the Republican-controlled Senate voted 67-29 in October 1998 to confirm Sotomayor's promotion to the appellate court.
The late Sen. Strom Thurmond, an Aiken Republican, voted against Sotomayor. Sen. Fritz Hollings, a Charleston Democrat, was one of four senators who chose not to vote.
House Majority Whip Jim Clyburn, a Columbia Democrat, called Sotomayor "a superb pick" by Obama to replace retiring Supreme Court Justice David Souter.
"The president has demonstrated that he is dedicated to nominating justices who have life experiences that will enable them to both sympathize and empathize with all Americans," Clyburn said.
Sotomayor, 54, grew up in a Bronx housing project with a single mom after her father died when she was 9.
Only the Senate will vote on Sotomayor's confirmation.
Oran Smith, head of the Palmetto Family Council, criticized Sotomayor as an activist judge.
"Judge Sotomayor's judicial record and writing clearly show she believes a court is not an applier of law written by elected representatives, but just another political advocate for social change," Smith said.