Romney, in a hastily arranged press conference in Jacksonville, Fla., said the Obama administration had made a “severe miscalculation” in its approach to the attacks in Libya and Egypt, and was sending “mixed signals” to the world by issuing a statement that he called “akin to an apology.”

Romney alleged that Obama was sympathizing with the attackers because of a statement that the US Embassy in Cairo issued condemning an online video offensive to Muslims as an example of religious intolerance. Romney said Wednesday the statement was issued after the attack in Cairo, as a response to it. But it was actually issued before the attack, specifically in response to growing anger in Egypt over the offensive video.

The day was one of the most dramatic in the presidential race, illustrating how rapidly the conversation could shift, and reflecting first reactions to a quickly evolving international crisis.

Obama did not mention his Republican rival in a Wednesday morning Rose Garden appearance condemning the attacks that killed four Americans, including the ambassador to Libya. But the president fired back at Romney on Wednesday afternoon in a prearranged television interview.

“There’s a broader lesson to be learned here: Governor Romney seems to have a tendency to shoot first and aim later,” Obama told CBS’s “60 Minutes.”

“And as president, one of the things I’ve learned is you can’t do that — it’s important for you to make sure that the statements that you make are backed up by the facts and that you’ve thought through the ramifications before you make them.”

The tenor of Romney’s critique — coming just hours after the deaths of four Americans on foreign soil — set off a vigorous debate over whether it was appropriately assertive, or cast Romney as out of his depth on foreign policy.

Critics said that in a moment that called for statesmanlike sobriety before all the facts are known, Romney went for partisan sniping and disregarded a long history of caution in the midst of unfolding foreign violence. But Romney stuck by his criticism throughout Wednesday, and expanded on a statement his campaign initially sent out late Tuesday night.

“I don’t feel that Mr. Romney has been doing himself any favors, say, in the past few hours, perhaps since last night,” Peggy Noonan, a conservative commentator and former speechwriter for President Reagan, said on Fox News. “Sometimes when really bad things happen, when hot things happen, cool words or no words is the way to go.”

But most Republicans were more tempered than Romney. House Speaker John A. Boehner did not criticize Obama, and Senator John McCain was complimentary of Secretary of State Hillary Clinton.

“It’s sad for the country that political ambition and expediency trumps decency and common sense,” Senator John Kerry, the Massachusetts Democrat and chairman of the Senate Committee on Foreign Relations, said in an interview with the Globe. “The interests of our country did not beg for a political fight with the loss of an ambassador and others in its first hours without even knowing the story. I find it very, very unfortunate.”

Romney’s press conference was sandwiched between one held by Clinton, who struck a somber tone reflecting on the life of a US ambassador, and Obama, who condemned the attacks.

“Since our founding, the United States has been a nation that respects all faiths. We reject all efforts to denigrate the religious beliefs of others,” Obama said. “But there is absolutely no justification to this type of senseless violence. None. The world must stand together to unequivocally reject these brutal acts.”

Romney’s morning press conference was quickly staged in Jacksonville, where he was planning to hold a rally with supporters.

Romney aides said they had been monitoring the events in the Middle East all day on Tuesday and by evening in the United States the situation was becoming clearer. Egyptian Islamists had scaled the US Embassy wall in Cairo, torn down the American flag, and burned it — all in a protest of an American-made film critical of Prophet Mohammad.