United States Constitution, Amendment 14
4. The validity of the public debt of the United States, authorized by law, [...] shall not be questioned.
Therefore, the debt ceiling is unconstitutional.
Congress is welcome to put restrictions upon what appropriations it can pass, such as pay-as-you-go budgeting rules. But, once a spending bill passes, the USA must pay back whatever money it borrowed as part of that legislation.
For example, if Congress and the President previously passed a "debt ceiling law" that capped federal debt at $12 trillion, but they also passed spending bills that increased the debt to $13 trillion, the USA must make good on the $13 trillion. There is no separate debt ceiling.
Government expenditures, i.e. the possibility of a federal government shutdown, are a different matter. If Congress and the President don't appropriate money by law, then they can't spend it.
The 14th Amendment, of course, came about in the aftermath of the Civil War, when the Union made good on its debts.
By the way, in a parallel universe in which George W. Bush was impeached, I would've loved it when the House tacked on an article of impeachment about Bush violating the 14th Amendment by calling the Social Security obligations a worthless bunch of IOU's.