No. Even if section 4 is interpreted as meaning that the the debts of the United States have first claim to Federal funds, all that would allow Obama to do (indeed he would be Constitutionally obligated to do this if that is how it is interpreted) would be to impound other appropriations and spend them on the public debt instead. In no event could section 4 be interpreted as the President has authority to incur debt not authorized by law above the debt ceiling set by law to pay interest on the existing debt.
So that could mean that Obama can impound funds appropriated for Utah, Alabama, Alaska, Louisiana, and other red states to pay the debt.
There aren't enough earmarks in the budget to do that. Besides, most such appropriations are authorized in a manner that he wouldn't be able to be that geographically selective. Impoundment is a blunt tool, and its use is regulated in part by the Impoundment Control Act of 1974.
As written, the Impoundment Control Act of 1974 (2 USC 681-688) requires Obama to send a message to Congress detailing which funds are to be impounded, and unless Congress passes a rescission bill (which has some special rules governing it for House and Senate debate) within 45 days, the money has to again be made available at that time. (Sooner if Congress passes an impoundment resolution telling the President to stop the impoundment.)
If the House wants to play hardball with the debt limit, impoundment is a tool Obama could use to say to Congress, you want the government to suddenly go to a balanced budget, here's what I'm cutting to do it. If nothing else, Obama could use impoundment to avoid a technical default on August 2, while negotiations drag on, but not without a lot of pain.