"The Speaker has and will lead the Republican conference to another majority in the 110th Congress," said a statement from Hastert's office. "Mark Foley has resigned his seat in dishonor and the criminal investigation of this matter will continue."
Meanwhile, Majority Leader John Boehner said the speaker had assured him months ago the matter had been taken care of. "It's in his corner, it's his responsibility," Boehner, R-Ohio, said in an interview on radio station WLW in Cincinnati.
Boehner, though, said in a letter to the Washington Times that he disagreed with their call for Hastert to resign.
"No one in the leadership, including Speaker Hastert, had any knowledge of the warped and sexually explicit instant messages" Foley allegedly sent to a teenage page.
Foley, a Florida Republican, resigned abruptly on Friday and has since checked himself into an alcohol rehabilitation program at an undisclosed location.
His departure left behind a trail of questions concerning the e-mails and instant messages he had sent pages over an unknown period of time. Beyond the details of his actions, Republican leaders feared the impact on the Nov. 7 elections, and the possible loss of their House majority.
One senior House Republican told CBS News correspondent Gloria Borger that this scandal "could be the congressional equivalent of Katrina. Our base is moral conservatives and we look like a bunch of hypocrites who just didn't want another scandal before the election."
The Washington Times, one of the most reliably conservative voices in the nation's capital, called for Hastert, R-Ill, to "resign his speakership at once" for not doing enough to investigate questions about Foley.
"Either he was grossly negligent for not taking the red flags fully into account and ordering a swift investigation, for not even remembering the order of events leading up to last week's revelations, or he deliberately looked the other way in hopes that a brewing scandal would simply blow away," The Times' editors wrote in Tuesday's editions.
"Mr. Hastert has forfeited the confidence of the public and his party, and he cannot preside over the necessary coming investigation, an investigation that must examine his own inept performance," The Times said.
Hastert and other leaders have laid out a complicated series of events. They say they first became aware of overly friendly e-mails from Foley to one underage male page about a year ago, but had no idea that the congressman had sent other, sexually explicit messages, to additional pages.
The speaker has said that Foley was confronted about his communications with the one page, and told to break off contact with him and all other pages.
Boehner, the second-ranking GOP leader, became the latest member of the party's high command to outline his involvement when he answered questions in his radio interview.
"I believe I talked to the speaker, and he told me it had been taken care of," he said, when asked about the e-mails that were not sexually explicit.
"In my position, it's in his corner, it's his responsibility. The clerk of the House, who runs the page program, the page board, all report to the speaker, and I believed it had been dealt with. Again, I didn't know the context of what even the original message (said)," he added.