It seems quite possible at this point that the ultimate result will be that the two Houses of Congress cannot agree on any legislation at all. If the WPR is applicable to this conflict, it would follow that our troops must be withdrawn from the hostilities. The WPR does not prohibit fighting if Congress disapproves it; rather, the WPR prohibits fighting (once its 60-day clock has run) unless Congress affirmatively approves it by passing a particular kind of legislation, the "specific statutory authorization."
Should the President be deemed to have the power to fight until he's forbidden to do so? Even if one reads the Constitution to give the President that much power, the War Powers Resolution does tell the President exactly that: stop fighting after 60 days unless we (Congress) authorize you to continue. So to uphold Obama's right to keep fighting -- assuming the WPR does apply to these events -- would one have to say that the WPR is unconstitutional?
The only other possibility I can see would be an argument that the WPR, though constitutional, has to an extent been superseded by subsequent congressional action. How's that? The idea would be that although the WPR rightly reflects the constitutional principle that both political branches must approve of our fighting an extended military engagement, there are many ways that Congress might express its approval. (I support reading the constitution to embody this principle, but certainly its validity is subject to debate; that, however, will have to be a topic for another day.) The WPR requires a "specific statutory authorization," but a subsequent Congress could choose to express its approval in some other way. One Congress can't, in general, tell a subsequent Congress how to meet its constitutional obligations. So perhaps Congress has in some other way given its approval, and also manifested its decision that it can give this approval without using the WPR's specified method.
There is at least some force to this argument. When Congress failed to enact a specific statutory authorization for the Yugoslavia bombing campaign, but did adopt a spending bill providing funding for that campaign, arguably its funding bill also constituted its expression of approval. Otherwise, if the constitution does require affirmative approval from Congress and if its funding bill wasn't that approval, then the funding bill financed an unapproved, hence unconstitutional, war -- and so it would seem that the funding bill was itself unconstitutional. Since there is a general presumption that Congress' statutes are constitutional, one should try to avoid an interpretation of its acts that renders them unconstitutional. So it might be argued that the spending bill passed for Yugoslavia silently, but by implication, repealed the WPR requirement of a "specific statutory authorization" for that war, a requirement that would have mandated that "any provision of law . . . including any provision in any appropriation Act" must in fact "specifically authorize the introduction of United States Armed Forces into hostilities . . . and stat[e] that it it is intended to constitute specific statutory authorization within the meaning of" the WPR. (Section 8(a)(1).)
Though all this might be right, it does not much help President Obama. Here there's been no spending law passed, nor any other affirmative expression of Congressional approval. So even if the "specific statutory authorization" requirement can be impliedly repealed for particular wars -- indeed, even if over the 40 years since the enactment of the WPR this requirement has somehow been impliedly repealed altogether (unlikely, since implied repeals are not favored when courts are asked to discern them in legislation) -- there's no evident direct approval of this war by any method.
Perhaps the only remaining possibility would be to discern such approval in actions Congress has taken that don't directly bear on the war. Congress has no doubt expressed its disapproval of human rights abuses, and Qadhafi surely was guilty of many of them. The UN Charter and the NATO Charter protect our security, and have been endorsed over and over through legislative action, and this fighting is authorized by NATO and by Resolution 1973 of the UN Security Council (though the scope of the Security Council's authorization has arguably been interpreted aggressively by the United States and NATO).
All this could be said -- but it all seems to miss the point. Grand principles and commitments may lead to war, but a decision must still be made to fight. I think that we are engaged in hostilities; the latest details to emerge, reported by Charlie Savage & Thom Shanker in the New York Times here, are that "[s]ince the United States handed control of the air war in Libya to NATO in early April, American warplanes have struck at Libyan air defenses about 60 times, and remotely operated drones have fired missiles at Libyan forces about 30 times, according to military officials." That's hardly constant bombardment, but it is persistent -- seemingly more than once a day on average. To share in making the decision to engage in such military action, I think Congress needs to actually focus on and decide it. Moreover, it's likely not enough that members of Congress support the decision in their private thoughts or in their speeches; they need to vote on it, up or down. That hasn't happened, and without it the fighting in Libya remains, I think, in violation of the WPR and the Constitution.