New York Mayor Bill de Blasio is facing a firestorm of criticism in the wake of the execution-style murder of two city police officers, as police union officials and ex-city leaders accuse him of stoking an atmosphere that made officers a target of "hate."

From his opposition to stop-and-frisk policies during his mayoral campaign to more recent comments about how he wanted his biracial son to be careful around police, the mayor's record on law enforcement is now front and center in an increasingly nasty war of words.

The fallout has even prompted a petition on Moveon.org calling for de Blasio's resignation -- so far, it has gathered nearly 60,000 signatures.

The accusations started flying almost immediately after authorities announced Saturday that officers Rafael Ramos and Wenjian Liu were fatally shot by a gunman who later killed himself. The suspect, 28-year-old Ismaaiyl Brinsley, had posted on Instagram that he was "putting wings on pigs" and suggested he was motivated by the deaths of Eric Garner and Michael Brown at the hands of police officers.

"There's blood on many hands," Patrolmen's Benevolent Association President Patrick Lynch said late Saturday. "... That blood on the hands starts on the steps of City Hall in the office of the mayor."

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Lynch and other officer organizations roundly condemned those who "incited" protests over the Garner and Brown deaths, and pointed the finger at the mayor's office. That night, a widely circulated video showed officers turning their backs on the mayor as he walked down a hall where the officers had been pronounced dead.

Some, including the mayor, have suggested the union criticism goes too far. In a statement, de Blasio's office reportedly said it is "unfortunate" that at this time of tragedy, "some would resort to irresponsible, overheated rhetoric that angers and divides people."

The temperature of the debate might be dialed down a bit in the coming days.

De Blasio, speaking at a police event Monday afternoon, urged all sides to "put aside" political debates and protests for the time being -- and focus on the families of the victims.

"These families are now our families, and we will stand by them," de Blasio said, calling the murders "an attack on every single New Yorker." He said the political debate will be dealt with "in due time."

Shortly afterward, at a press conference alongside de Blasio, NYPD Commissioner William Bratton said he spoke with leadership of all the police unions, and claimed they have agreed on "standing down" until after the funerals. "I want to thank them for that action," he said.

On the night of the murders, de Blasio spoke in support of the police force and called the murder "an attack on everything we hold dear."

But the murder of Ramos and Liu nevertheless has blown open a rift with the police force unlikely to heal soon. The police unions already were at odds with de Blasio over, among other issues, his handling of the response to the Garner case -- Garner died in a struggle with police after being approached for selling loose cigarettes, but no officer was charged.

The de Blasio administration allowed days of demonstrations and drew criticism from union leaders for his comments during that period -- including saying his family had to "train" their biracial son how to act around police.

"I don't put the blood of these police officers on the mayor's doorstep," former New York City Mayor Rudy Giuliani told Fox News on Monday. "I lost police officers, Bloomberg lost police officers. What I do hold him responsible for is letting those demonstrations get out of control. ... He's guilty of creating an atmosphere of police hatred in certain communities."

He accused de Blasio, President Obama and others of fueling an "atmosphere of hate" toward officers.

Deputy U.S. Attorney General James Cole on Monday rejected such criticism.

"We have supported our police officers and our law enforcement personnel from the time I have gotten here," Cole said.

According to The New York Times, even before the murder of the officers, union leaders sent around a letter letting officers request de Blasio not attend their funerals if they died in the line of duty.

Outside the union ranks, several former top-ranking New York officials -- including Giuliani -- have started putting pressure on de Blasio.

Former Gov. George Pataki tweeted: "Sickened by these barbaric acts, which sadly are a predictable outcome of divisive anti-cop rhetoric of #ericholder & #mayordeblasio. #NYPD"

Former NYPD Commissioner Ray Kelly told ABC's "This Week" that de Blasio ran an "anti-police campaign" last year, and said his comments about having to train his son "set off this latest firestorm" with the officers.

Fox News' Lucas Tomlinson and The Associated Press contributed to this report.