WASHINGTON  President Hosni Mubarak’s refusal to step down on Thursday, after a day of rumors galvanized the crowds in Cairo, confronts the Obama administration with a stark choice: break decisively with Mr. Mubarak or stick to its call for an “orderly transition” that may no longer be tenable.

On a day of dashed hopes in Egypt, the administration’s attempts to balance the democratic aspirations of the protesters against a fear of contributing to broader instability in the Middle East collided head-on with Mr. Mubarak’s defiant refusal to relinquish his office.

To some extent, Mr. Mubarak opened the door for President Obama to appeal even more directly to the protesters, some of whom have felt betrayed by the administration’s cautious approach, saying it placed strategic interests ahead of democratic values. In his speech, Mr. Mubarak said he would not brook foreign interference, suggesting that he was digging in his heels after days of prodding by the United States for “immediate, irreversible” change.

Mr. Obama’s remarks earlier in the day, in which he celebrated the hopes of a “young generation” of Egyptians, were broadcast in Cairo, drawing cheers from the protesters.