Republicans interspersed their criticism of Holder about the leak probes with criticism of his failure to turn over more documents to a House committee investigating a flawed gun-smuggling probe in Arizona, Operation Fast and Furious.
Regarding Fast and Furious, Cornyn called on Holder to resign. He said the attorney general had misled Congress in February by embracing a letter to Congress denying there were problems. Hundreds of illicitly purchased weapons wound up south of the border, many of them at crime scenes.
Holder offered to sit down with Republicans and reach an accommodation to avoid what the attorney general characterized as an impending constitutional crisis over Justice Department documents the attorney general says involve highly sensitive law enforcement investigations and prosecutions.
Republicans on the House Oversight and Government Reform Committee have scheduled a June 20 vote on a proposed contempt of Congress citation directed at Holder for not turning over the material.
On the Senate floor, Graham said that not appointing a special counsel would set a "precedent that will haunt the country and this body and future White Houses in a way that I think is very disturbing."
McCain's move to get swift Senate passage of his resolution was blocked by Democratic Sen. Ron Wyden of Oregon, who said the effort was premature.