"I, Barack Hussein Obama, do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States."
The Constitution tells us that to faithfully execute the office the President shall:
from time to time give to the Congress information of the state of the union, and recommend to their consideration such measures as he shall judge necessary and expedient;
he may, on extraordinary occasions, convene both Houses, or either of them, and in case of disagreement between them, with respect to the time of adjournment, he may adjourn them to such time as he shall think proper;
he shall receive ambassadors and other public ministers;
he shall take care that the laws be faithfully executed, and shall commission all the officers of the United States.
The appropriations are lawfully commanded by Congress--to fail to spend them is, since the budget reforms of the 1970s, to break the law--the repayment of the debt is mandated by the Constitution's Article XIV, as is the limitation of the government's authority to take property via taxation or otherwise to the amount lawfully commanded by Congress.
There are two ways in the absence of a debt-ceiling increase for the President not to break the law:
Find some other debt of the U.S. government--like, as Bob Rubin did, the debt of the U.S. government to the Federal Employees' Thrift Savings Plan--where the Trustee will not complain if the government does not pay its debts for a while, and then have the government not pay its debts for a while.
Mint the damned coins already.
(2) look strongly preferable to me--the exercise of the government's not paying its debts to people who can't gain standing to sue and then saying that unpaid debts by the government are not part of the debt subject to limit has always seemed ugly.