WASHINGTON — Speaker John A. Boehner stood his ground on Sunday alongside the most conservative Republicans in Congress, insisting that the House would not vote to finance and reopen the government or raise the nation’s borrowing limit without concessions from President Obama on the health care law.

“The fact is, this fight was going to come one way or the other,” Mr. Boehner said on the ABC News program “This Week,” adding, “We’re in the fight.”

With his hard line, Mr. Boehner reaffirmed that the stalemate with the White House over the six-day-old government shutdown was now compounded by an even more economically risky fight over raising the government’s borrowing limit by Oct. 17 to pay for bills already incurred.

Most of the government remains shuttered with no end in sight, and markets and businesses are growing increasingly fretful over the chaos that could result from the first government default on its debt.