The officer, Charlie Rispoli, a 14-year veteran of the police department in Gretna, referred to Ms Ocasio-Cortez in a post on Thursday as “this vile idiot".
The post continued, saying she “needs a round – and I don’t mean the kind she used to serve”, a reference to her past work as a bartender, according to a screenshot of his comment obtained by nola.com, which covers New Orleans.
He was responding to a post by the satirical website TatersGonnaTate with the headline “Ocasio-Cortez on the Budget: ‘We Pay Soldiers Too Much'".
Wayne Rau, a councillor and the current mayor of Gretna, a city of about 18,000 people across the Mississippi River from New Orleans, condemned Mr Rispoli’s post in an interview on Sunday.
“I’m pretty appalled by it,” Mr Rau said. “I don’t think this is a representation of the police department of the city of Gretna.”
Asked whether Mr Rispoli’s Facebook post was a threat, he said, “I’m not sure how to take that post.” He said it should be up to the city’s police chief, Arthur Lawson, to decide on any disciplinary or criminal action.
Three of Mr Rau's four other members in Gretna City Council were not immediately available for comment, while the fourth, Jackie Berthelot, directed questions to the police department.
Efforts on Sunday to reach Mr Rispoli, who appeared to have deactivated his Facebook account, were also unsuccessful. Corbin Trent, a spokesman for Ms Ocasio-Cortez, declined to comment. Mr Lawson, who is an elected official, did not immediately respond to a request for comment.
The chief said he was disturbed by the post. “I will tell you this: This will not go unchecked,” he said.
“I’m not going to take this lightly and this will be dealt with on our end. It’s not something we want someone that’s affiliated with our department to make these types of statements. That’s not going to happen.”
The chief told the website on Friday that the post appeared to violate the department’s social media policy but he did not think it rose to the level of being a threat.
Ms Ocasio-Cortez is part of a quartet of first-year Democrats in the House who have received death threats and were told by Donald Trump to “go back where you came from” in a racially charged post on Twitter.
Belinda Constant, the mayor of Gretna, said she had been out of the state and was not immediately prepared to comment. “I will be looking into the situation,” Ms Constant, a Democrat, said.
Eva Malecki, a spokeswoman for the US Capitol Police, declined to say whether the agency was investigating.
“The mission of the United States Capitol Police is to protect and serve Congress,” she said. “However, we do not discuss how we carry out our protective responsibilities.”
The four congresswomen were also the subject of a recent Facebook post by the Republican County Chairmen’s Association of Illinois, which depicted them in a mock movie poster that referred to them as the “jihad squad".
“Political Jihad is their game,” said the post, which was published on Friday and later deleted. “If you don’t agree with their socialist ideology you’re racist.”
It showed Ilhan Omar of Minnesota and Ayanna Pressley of Massachusetts brandishing guns.
On Sunday night, the state’s Republican Party chairman, Tim Schneider, issued a statement disavowing the post, which he said did not reflect the party’s values.
“Bigoted rhetoric greatly distracts from legitimate and important policy debates and further divides our nation,” Mr Schneider said, adding, “I urge everyone who opposes them to keep the rhetoric focused on policy and ideology.”
The Washington Post