The Washington Times has accused Hastert of barely pursuing warnings about former representative Mark Foley's overtly sexual internet messages to teenage boys and said in an editorial that he "must do the only right thing, and resign his speakership at once".
Hastert has denied any knowledge of Foley's messages until they were made public on Friday, and he rejected calls for his resignation.
He said, "I'm not going to do that", and said he was trying "to do the right thing".
Hastert said he had talked to more than 200 House members and the idea of his resignation did not come up, but rather "people urged us to go forward".
He said: "I'm running for speaker and I would be the candidate if our party won, I suspect. We are trying to turn this thing around."
George Bush, the US president, did not respond to questions in California over whether Hastert should resign, but said he was "confident he will provide whatever leadership he can to law enforcement" in its investigation of Foley.
"He's a father, teacher, coach who cares about the children of this country," said Bush, who said he was "dismayed and shocked" by Foley's behaviour.
The scandal widened as ABC television revealed salacious new details of additional emails which suggested a possible relationship with an underage congressional aide.
In the email exchanges published by ABC, Foley and the young congressional page traded internet kisses and made plans for a future encounter.
Until Tuesday, most of the exchanges made public showed Foley propositioning congressional pages, who did not appear entirely to return his advances.
The scandal is a blow to Republicans and provides fresh ammunition to Democrats before congressional elections on November 7, when Democrats must pick up 15 House seats and six senate seats to claim a majority in each chamber.
Foley's lawyer said the six-term House veteran is an alcoholic gay man who was molested by a clergyman when he was a teenager.
"Any suggestion that Mark Foley is a paedophile is false, categorically false," the attorney, David Roth, told a news conference in Florida.
Foley is at an undisclosed alcohol rehabilitation centre and his treatment will last at least 30 days, Roth said.
Members of both parties, including some leading conservatives, have said that any congressional leader who had known the content of Foley's messages and had failed to take action should step down.