There's a substantive debate going on among legal scholars about how far Bush can take the executive privilege claim, and whether, in the end, he'll be caught Congress's (and the law's) net.
Some excerpts:
The heart of the House case is... : Even if the privilege were properly raised and applicable, it would be outweighed by the House's need for information relevant to investigating serious wrongdoing. As in the Nixon case, so too here, there is no plausible national security justification for keeping the material secret, and prior administrations have declined to assert executive privilege where Congress sought evidence of wrongdoing by the administration itself. Whether this objection is correct as a matter of case law depends on whether Nixon—with its demanding burden of persuasion on the administration—applies outside the context of a criminal prosecution. The Cheney case suggests that it may not, but this is an open question: We can grant that executive privilege is entitled to greater protection in civil cases than in criminal cases; it does not follow that it is entitled to less protection in a direct conflict between the House and the President. (Michael Dorf, Columbia)
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The decision to fire the U.S. Attorneys was, in the President's own words, made by "the White House," after consultation with DOJ, but without the President's own involvement. No one is arguing that the privilege applies to the internal DOJ communications, or to the communications between DOJ and White House officials (many of which the Administration has provided to Congress). Indeed, if the decision to fire the officials had been made by the Attorney General, there would be no privilege claim. Why, then, should the constitutional question change merely because the ultimate decisions to remove the officials, and most or all of the pressure to affect criminal prosecutions and investigations, were the work of (I'm surmising here) Karl Rove and Harriet Miers, rather than Alberto Gonzales? In recent decades, more and more functions of the Executive branch have been shifted from agencies to the burgeoning White House staff. Congress already has less ability to oversee those functions, because most of the those White House officials are not Senate-confirmed and are not subject to ordinary oversight. To give those White House officials an additional constitutional privilege that would not be enjoyed by DOJ officials engaged in the same functions seems rather odd. (Marty Lederman, Georgetown)
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The most important thing Congress could do, if it genuinely believes there is evidence of criminal wrongdoing by persons in the executive branch, is to call for the appointment of a special prosecutor. The Independent Counsel Act has, thankfully, gone to a well deserved repose. But special prosecutors are still quite possible and constitutional, and they may be entirely appropriate given the fact that large swaths of the Justice Department are probably conflicted out of investigating these matters.
To be sure, the President will probably refuse to appoint a special prosecutor if he can help it. After all, the last one he appointed produced Scooter Libby's conviction. That is where Congress comes in. It has many different means at its disposal to pressure the President, including making the option salient before the public, raising public ire against the President for refusing to appoint a special prosecutor, refusing to move on nominations and appointments that the President seeks (or to pass various appropriations bills he likes) until he appoints a special prosecutor, and so on. The real question is whether there is the political will in Congress to do this. If not, I suspect that the various proceedings will drag on until the President leaves office. At the moment, that seems to be his preferred outcome. (Jack Balkin, Chicago)