Referring to Saturday’s “Fox & Friends” segment, she said, “A host on Fox & Friends wrongly states that, “al-Baghdadi was able to sneak away under the cover of darkness after a New York Times story” and that the U.S. government “would have had al-Baghdadi based on the intelligence that we had except someone leaked information to the failing New York Times.”

On Sunday, The Times also published a fact-check of the Fox News article and the president’s tweet.

On Monday, Ms. Rhoades Ha said in an email that the segment that aired earlier that day “wasn’t an apology, nor did it begin to address the larger issues with the Fox & Friends Weekend segment, one of which was sheer hypocrisy.”

She continued, “The host railed against The New York Times for covering a raid stating that the U.S. government ‘would have had al-Baghdadi based on the intelligence that we had except someone leaked’ to The New York Times when Fox News had covered the same raid three weeks earlier in a segment in which their correspondent said, ‘The newly recovered intelligence may bring U.S. closer to Baghdadi’s kill or capture.’”

“According to the curious logic of the Fox & Friends host, Fox News itself was unpatriotic,” she added.

The updated segment on “Fox & Friends” on Monday morning consisted of airing General Thomas’s comments again, and adding about 15 seconds dedicated to The Times’s response. It did not respond directly to the paper’s complaints.

Fox News did not respond directly Monday morning to questions about whether any further correction or clarification was planned, sending links to the “Fox & Friends” segment as well as the relevant portion of the show’s transcript.

Later in the day, the network sent a statement that said that “neither Fox News’ report nor the subsequent on-air coverage was inaccurate.’’ The statement said the network found it “beyond disappointing’’ that The Times “decided to blame Fox News for comments made publicly by General Thomas.’’

The Times’s full letter to Fox News is below: